PHYS 110 LAB Manual-Updated - Tagged

INTRODUCTION How to Succeed in Physics Lab Read the lab manual before coming to class to become familiar with the experiment. Lecture and Lab are NOT in perfect synch, so you may have to give the textbook a look also. You should take responsibility to learn safe operating procedures from the lab instructor. The lab manual is also occasionally a good source of safety tips. With electrical circuits, no power is to be supplied unless OK'd by instructor or lab tech. Report any accidents immediately! You will work with a lab partner to take data, but you are individually responsible for your own data. All subsequent calculations, graphs, etc. are also your own individual responsibility. Original data MUST be in ink. If you change your mind, cross out with a single stroke, and enter new datum nearby. Do not leave the lab room without obtaining the instructor's signature on your original data sheet. Without it, your lab report will not be accepted. No exceptions. The lab has been designed to be a "low pressure" experience. We hope it is an enjoyable one as you take the time to become familiar with new equipment and experiences. Still, you should aim to complete all data-taking, all necessary calculations, reach all conclusions, and at least sketch all graphs before you leave. It's well known (to those who know it well) that once you walk out that door, all work on lab reports will take longer. Besides, most of the grade for the course will come from the lecture part, so spend your time accordingly. Before taking good data, run through the experiment once or twice to see how it goes. It is often good technique to sketch data as you go along, whenever appropriate. The Report: Your Lab Report should be self-contained: It should still make sense to you when you pass it on to your grandchildren. It should include: a) Front page: your original data sheet with your name, partners and date. The original data MUST be in ink. Report not acceptable if original data is in pencil or if data sheet was not signed by your instructor. (So... don't leave lab room without it.) b) Additional pages with data and calculations in neat tabular form. If the original came out messy, you should rewrite your data before continuing with calculations. c) Any graphs. Neatness counts! It's one of the aims of this lab to produce students that know how to produce a decent graph. d) Answers to any Questions 3

number of digits in which any confidence can be placed was three, two, and four, respectively. Another way of representing the error is to explicitly indicate it. Suppose that we measure a length, which we find to be 1.520 m, and we believe that this measurement is accurate to within a centimeter. We could then write the result of our measurement as 1.520±0.005 m, which means that our result is between 1.515 m and 1.525 m. In general, if we are measuring a quantity x, we can write the result of a measurement of x as x best ±Δx, where x best is our best estimate of the quantity being measured, and Δx, which is a positive number, is the value of the uncertainty in our measurement. We can also use what is called the fractional error. This is just Δx /| x best |. To get the percent error, we just multiply the fractional error by 100. In our example, we have that x best =1.52 m, Δx =0.005 m, the fractional error is 0.0032 and the percent error is 0.32. Note that the fractional error and the percent error have no units. Finding the uncertainty We have been discussing representing the result of a measurement as x best ±Δx, but given a set of readings from an instrument, how do we find x best and Δx? We find x best by taking the average of the readings, so if we have made three measurements of a length, and have found the results 1.26 m, 1.28 m and 1.25 m, we have that x best = (1.26 m + 1.28 m +1.25 m )/3 = 1.26 m. Finding Δx is a bit trickier. First, it cannot be any smaller than the precision of our measuring instrument. If we are using a meter stick to measure a length, and the smallest units marked on the meter stick are millimeters, then the uncertainty in our measurements will be approximately one millimeter (it can actually be a bit less than this, because we can estimate the length to a fraction of a millimeter – this would give us an uncertainty of about 0.3 mm). However, it might be bigger due to random errors. In order to do a detailed analysis of random errors, it is necessary to know a lot more about probability theory and statistics than you do now, so we will just give you the answer. Suppose you have made N measurements and the results are x 1 , x 2 , …x N . You first calculate the average value of the measurements, x best , and then you use it to find what is called the standard error, s, which is given by s = [(x 1 - x best ) 2 +(x 2 - x best ) 2 +…(x N - x best ) 2 ] 1/2 /[N(N-1)] 1/2 . The uncertainty, Δx, will just be the sum of the uncertainty due to the measuring instrument and the standard error, s. Computations using raw data How do you combine your carefully gathered data with other numbers in an expression? With a little common sense, and a hand calculator, you can verify that the following rules should be followed: Addition and Subtraction: Suppose we want to add two measured lengths, 0.36±0.01 m and 0.48±0.02 m. Why would we want to do this? Each length might be the length of a block, and then the sum would correspond to the length of the object we get when we put the two blocks together. What can we say about the length of the combined blocks? Our best estimate of their combined length is just 0.36 m + 0.48 m = 0.84 m, but we have to worry about the uncertainties too. The longest the combined blocks can be is (0.36 m + 0.01 m)+(0.48 m 6

TUTORIAL # 2 Making a Good Graph by Hand Start thinking about a nice title. e.g., " The Square of the Period (T 2 ) of a Simple Pendulum vs. its Length (L)" [ A shorter title would have been even better] Keep your axes straight: If you need to plot "A vs B", or "A as a function of B", then A is on the vertical axis and B is on the horizontal axis. vertical axis = y-axis = the "ordinate" horizontal axis = x-axis = the "abscissa" The crucial part is choosing the range and scale for each axis. Two examples: a) 0 to 5 sec; 5 graph boxes =1 sec. b) -300 to +200 degrees; 2 boxes =100 degrees The range must be: just large enough to accommodate all your data, yet allow a readable scale. The scale should be: spread out enough so your data take up most of the graph area, and labeled so plotting (and reading) is easy. Label the x and y axis with the appropriate magnitudes. These should be round numbers which cover the entire range of values that you will be plotting. A common mistake is to label too many boxes. If you are trying to show that one quantity is proportional to another, or if you are not told otherwise, zero is part of the range and must be located at the Origin. The numbers should be evenly spaced with the same number of boxes between the same increase in numbers including the space from zero to the first non-zero number. Choose an appropriate number of boxes between numbers. It is better to have 5 boxes between numbers than 4 since it is easier to interpolate in the first case than in the second. (Similarly 10 is better than 8, and 2 is better than 3.) Plot the results of your measurements on this graph. Where appropriate, you should include error bars to indicate the uncertainty in your measurements. These error bars are not of arbitrary size but should be of the size of your uncertainty on the scale dictated by the numbers on the axis of your graph. When you draw the line that best fits your data, the line should be a smooth one that need not go through any points. In general, there should be as many points on one side of the line as on the other. If you have done your work properly, the line should pass inside of the error bars for each point. (If it does not, that may be an indication that there is something wrong with the point in question. Perhaps you miss-recorded a measurement, or your estimate of the error was too small, or there was something wrong with the 9

LABORATORY EXERCISE #1 Density; Significant Figures Objectives To become familiar with instruments to measure length (meter stick, vernier caliper, micrometer) and to appreciate the difference between accuracy and precision of experimental measurements. Incidentally, to learn the concept of mass density, and practice proper use of significant figures. Equipment and Supplies Metal samples, electronic balance, meter sticks (long and short), vernier caliper, vernier model, micrometer calipers. Discussion The mass density of a body measures the amount of mass per unit volume of that body. Definition: Density of a body = (body's mass)/(body's volume) or D = M / V The density of a substance does not depend on the shape or the particular amount of that substance. For example, the density of a small gold ring and the density of a large gold brick should be the same number, i. e., the density of gold. As part of your density calculations you will need to compute the volume of a regular solid from its linear dimensions. The quality of your volume measurements will thus depend on how precisely you can measure lengths. You will use three progressively more precise instruments: a wooden "meter stick", vernier calipers and a micrometer. Your instructor will demonstrate how to use each one. In each case, you should read the instrument to the smallest division plus one more digit by estimation. So, a meter stick with millimeter markings can be used to estimate a length to the nearest tenth of a millimeter, e.g., 12.4 mm, or 1.24 cm. This estimate by "eye" is often only good to ±2 or 3, as you can verify by repeating the measurement or asking your partner to do the estimating. A vernier is an invention that removes the uncertainty in reading to the nearest tenth between adjacent markings, thus increasing the precision of the final measurement. Your instructor will demonstrate how this is done with the vernier model, prominently displayed in the front of the laboratory room. On Errors and Uncertainty in a measurement: When you work out a math computation, the numbers are usually considered exact, e.g., 1.1 x 1.2 x 1.3 = 1.716. But when a number represents a physical measurement, it is never exact because of the limitations of the instrument used, or the way it was employed, etc. It is essential, therefore, that each experimental result be presented in a way that indicates its reliability. A very simple way to do this is by the use of significant figures. (See Tutorial #1, on Significant Figures) 12

LABORATORY EXERCISE #1 Data Sheet: Density; Significant Figures Date:_________ Name:______________________________ Partners:_________________ Instructor's signature:____________________ A. I have measured the volume of the air in this laboratory room and hereby report that using an assumed density of ____________ kg/m 3 , I find that the room contained _________ kg of air. {How close you come to the "true" answer is how accurate your measurement was.} I certify that, should another careful experimentalist attempt to make the same measurement, she (he) will come within * ± ________%, approximately, of my value. {Too high a number here, will show you to be a sloppy experimentalist, unable to squeeze the best from the tools available. Too low a number will expose you as a fraud, claiming a precision unwarranted by your instruments and methods of using them.} ________________________ (signed) B. For each of the three methods in part B, compute the density of the metal block to the appropriate number of significant figures. Present data, calculations and answers below. Use additional sheets if necessary. * 15

Laboratory Exercise #2 Position and Velocity With an Introduction to Graphical Analysis Software Objectives: To understand the relation between position and velocity, and to understand how to represent each of them graphically; To familiarize with the tools used in mechanics experiments. Equipment and supplies: Sensor cart (green), track with end stop, computer, Graphical Analysis software. Computer and software In this semester we will do a series of experiments to explore kinematics, dynamics, momentum, and energy using one or two sensor carts on a track. The data of motion will be collected by wireless sensors and analyzed by a computer. We will first learn how to connect the sensors to your computer. A. How to connect the cart sensors to the computer. 1. Place the green sensor cart on the track. The arrow should points to the end stop. Disconnect the charging cable from the cart if it has not yet been disconnected. 2. Open Graphical Analysis program. When the window opens up, a. Click “Sensor Data Collection”. b. Select the sensor number that matches the number on your green cart. Then click “Done”. Two graphs, the Position graph and Velocity Graph, will appear on the screen. (If you do not see the correct sensor number, press the switch on the side of the cart to turn it on.) c. We will need only one plot only. To do this, click “View Options” at the upper right corner. Choose “One Graph”. d. Set duration of data collection Click “Mode” on the lower left corner to open the Time Based window. Under “End Collection”, type in 15 seconds. Then click “Done”. This will set the total time for data collection to 15 seconds. e. Zeroing the position axis Move the cart to the starting point at the opposite end of the track from the end stop. Click “Position” on the right lower corner and click zero. This will make current position on the track to be position zero. B. Plotting Position-Time Graph with hands. 1. Starting at the far end of the track from the end stop use your hand to move the cart slowly and steadily toward the end stop. Practice this motion and try to complete the motion through the track in approximately 15 seconds. Sketch the position-time graph (1) for this motion in the report sheet and label the axes with appropriate scales and units. 18

b. Next find the average velocity from the graph. This can be done by recording, from the position graph, time t 1 and t 2 when the cart passes the red marks on the track and find the average velocity during this period from the Velocity graph using View Statistics Function. Compare the velocity measured manually in (a) with the average velocity reading from the graph. Present a screenshot of the graph using a mobile phone. E. Verifying the equation x 2 = x 1 + v ( t 2 − t 1 ) 7. Suppose we want to find position information from a velocity graph. If an object is moving with a constant velocity, then the distance it travels between two times, t 1 and t 2 , is just the velocity measured at t 1 multiplied by (t 2 -t 1 ). We can see if our graphs confirm this. Set the cart in motion and start collecting data. Zoom in all graphs. Take screenshot. Choose two times in the region where the velocity is nearly constant, which we shall call t 1 and t 2 , and find the positions corresponding to them using the position-time graph. Let us call these positions x 1 and x 2 . Read the velocity at t 1 . Calculate x 2 . Compare the calculated x 2 with the measured value. 8. Repeat Step 7 for 2 more pairs of t 1 and t 2. At the end of this experiment, re-connect the charging cable to the cart and place the cart on the track. 21

Laboratory Exercise #2 Position and Velocity Date ______________ NAME:______________________________Partners:________________________ Position-Time Graphs (1) Slow and steady motion to the end stop (2) Faster and steady motion to the end stop (3) Slow and steady moving away from end stop (4) Faster and steady moving away from end stop (5) Prediction 22

(6) Screenshot of position- time graph which corresponds to the prediction in (5). (7) Paste the screenshot that reproduces the Position-Time graph shown in Section 7. 23

Velocity-Time Graphs (a) Slow and steady moving toward end stop (b) Faster and steady moving toward end stop (c) Slow and steady moving away from end stop (d) Faster and steady moving away from end stop 24

D Position and Velocity Graphs Provide a screenshot for position and velocity graphs obtained in D. Data from the position-time graph: Record five points (with units) from the position-time graph, calculate the average value. Position, x Time, t Velocity = Δx Δt (slope) Average velocity = Comparing the average velocity with the average velocity reading from the View Statistics function. 25

D 6 Compare manual and automated measurements Positions of red marks X 1 =___________________. X 2 =__________________________ Stopwatch reading: ____________________________________ Average Velocity=_____________________________________ Provide a screenshot of the position- and velocity-graphs. Deduce the average velocity from the graph and compare with the average velocity obtained manually. 26

Verifying the equation x 2 = x 1 + v ( t 2 − t 1 ) Paste the screenshot to be used for the analysis. Use the position, x 1 , corresponding velocity, v , at a given time, t 1 , to predict the position x 2 at time t 2 using the equation x 2 = x 1 + v ( t 2 − t 1 ) . Find the x 2 values for three different t 2 . Compare the predicted x 2 with the readings from the position-time graph. If the predicted values do not match the experimental values, explain why. 27

LABORATORY EXERCISE #3 Acceleration Objectives: To study the relation between position, velocity, and acceleration when the acceleration is constant. Equipment: Sensor cart, track with end stop, 10-g, 20-g cylindrical masses, pulley, strings. Introduction: Velocity tells us how position changes with time, and acceleration tells us how velocity changes with time. If an object, like a cart, has a constant acceleration, then we can find what the acceleration is by calculating Δ v / Δt , where the velocity of the object changes by an amount Δ v in a time interval Δ t . The relation between acceleration and position is more complicated. In particular, the position of an object with constant acceleration depends not only on the acceleration, a , but on the initial velocity, v 0 and initial position, x 0 , of the object as well. The exact relation is x ( t )= 1 2 at 2 + v 0 t + x 0 . A. Velocity and Acceleration Our first task is to understand the relation between velocity and acceleration. We will again use the cart on a track, but this time there will be weight attached to the cart to make it accelerate. First we will look at the velocity and acceleration of the cart when it is speeding up, and then we will look at them when the cart is slowing down. 1. Set up the cart on the track with the pulley attached to the end of the track. The 20- gram weight is then attached to the cart through a string which is laid over the pulley. 2. Open Graphical Analysis software. Connect the sensor to the computer. You should see two graphs on the screen. Change the vertical labels to velocity-time graph, acceleration-time graph. 3. What you are going to do is to move the cart to the far end of the track with the weight lifted to the height of the pulley, zero the position, click Collect and release the cart. The position sensor will collect data while the cart is moving. Before you do this, sketch in graphs (1) and (2) a picture of what you think the velocity-time graph and acceleration- time graphs will look like (hint: the weight gives the cart a constant acceleration). Now do it and see how the actual graph and your prediction compare. When you do the experiment, hold on to the cart until you are ready to take the data, and then release it (do not give it a push). Catch the cart before it hits the pulley. 4. Read off four pairs of points from the velocity-time graph. Calculate 28

( Δ v / Δt ¿ =( v 2 − v 2 )/( t 2 − t 2 ) for each pair, and find their average value. Now use the View Statistics feature to read average acceleration and compare to the computed average values in the corresponding time range. 5 Deceleration (negative acceleration) a. Now we want to try something a bit different. You are now going to start the cart with the 10g weight laying over the pulley and hanging in the air, give the cart a gentle push away from the pulley. (Pushing too hard can result in the weight hitting the pulley and make the analyze difficult.) The cart will move away from the pulley, decelerate, turn around and come back. Before you do this, sketch a prediction for the velocity-time and acceleration-time plots in graphs (3) and (4). Pay particular attention to what you think is going to happen to the velocity and the acceleration at the turning point. b. Before taking the data, we need to re-define the positive direction because the initial velocity of the cart is in the opposite direction of the arrow. Click Position on the lower right corner of screen, enable Reverse. Then Click Collect. Repeat this procedure if necessary until the velocity-time graph has at least two seconds of straight line. Click the magnifier to zoom in. Take a screenshot. B. Position, Velocity and Acceleration We now want to understand how position is related to acceleration and velocity when there is a constant acceleration. 1. Click View Options to choose 3 Graphs. Change the vertical labels to display Position-, Velocity, and Acceleration graphs. From class we know that the velocity will be a linear function of time, ( t )= At + B , and the position should be a quadratic function of time, x ( t )= At 2 + Bt + C . We want to figure out what those functions are for the data in our last run. The coefficients that appear in those functions are related to the acceleration, initial velocity ( at t=0) and the initial position (at t=0). 2. Choose a segment in the velocity-time graph, which should look something like a straight line, Apply Curve Fit (under Graph Options). What physical quantities do A and B represent? Is either one related to your acceleration graph? If so, compare the value you found for A or B to the relevant value on the acceleration graph. 3. Now go to the position-time graph. We want to find an equation for this curve, which should be something like a parabola. This means we should have, x ( t )= At 2 + Bt + C . We want to find the coefficients A, B, and C. We can do this by using the Quadratic Fit under View Statistics Leave the fitting curves on the screen and take a screenshot. Read off the values of A, B, and C. What physical quantities do A, B, and C represent? Are any of them related to either your acceleration graph or to your velocity-time graph? If so, compare the values you found here to the values you found from your acceleration and velocity-time graphs. 29

LABORATORY EXERCISE #3 Acceleration Date ______________ NAME:______________________________Partners:________________________ Instructor's Signature:____________________ Velocity and acceleration Predict the velocity-time and acceleration-time relations when the cart is pulled by the weight. (1) (2) Paste the screenshot for the velocity and acceleration graphs for empty cart. 30

Read off four pairs of points from the velocity-time graph. Calculate Δ v Δt = v 2 − v 1 ❑ t 2 − t 1 t for each pair, Velocity , v Time, t Slope a = Δv Δt Average value= Average of four acceleration values from the acceleration-time graph = How does the values of average acceleration from the acceleration-time graph compared to the average values of ( Δ v / Δt ) ? 31

A 5 Sketch predicted velocity-time and acceleration-time graphs for the procedures outlined in step (5). (3) (4) Paste the screenshot obtained in 5 b with the cart pulled by 10g over the pulley. Compare the measured with the predicted graphs. Use your own words to explain why the graphs make sense. Use Apply Curve Fit over a linear segment in the velocity-time curve with the equation v=At+B and find A and B . What physical quantities do A and B represent? Is either one related to your acceleration graph? If so, compare the value you found for A or B to the relevant value on the acceleration graph. What are the physical meanings of A and B. 32

B. Paste the 3 graphs screenshot here. Fit the position-time curve with x ( t )= At 2 + Bt + C and read off the values of A, B, and C. What physical quantities do A, B, and C represent? Are any of them related to either your acceleration graph or to your velocity-time graph? If so, compare the values you found here to the values you found from your acceleration and velocity-time graphs. 33

LABORATORY EXERCISE #4 Vectors Objectives: To learn adding vectors by their components and to compare the calculation with an experiment. Equipment and Supplies Force table, three low-friction pulleys, mass hangers, mass set. Discussion Consider a vector A that lies in a plane. It can be expressed as the sum of two components. The components are usually chosen to be along two perpendicular directions. An example is shown in Figure 1. The vector A can be resolved in to its x - and y -components by drawing two lines perpendicular to the x - and y -axes. Then A x and A y are the x and y components of A , respectively. From Figure 1, A x = A cos  and A y =Asin  . Figure 1 Consider then the addition of two vectors A and B to give a resultant vector C , as shown in Figure 2. 34

Figure 2 C = A + B (1) The x and y components of these three vectors satisfy the following relations: C x = A x + B x (2) C y = A y + B y (3) where A x = A cos  a (4) A y =Asin  a (5) B x = B cos  b (6) B y =B sin  b (7) The magnitude and angle of C can be determined by: C=( C x 2 + C y 2 ) 1/2 (8)  c = tan -1 (C y / C x ) (9) In the present experiment, the methods of vector addition is used to study the condition of equilibrium for three forces on a force table. These forces are created by the weights attached to the mass hangers. The diagram of the vectors involved is illustrated shown in Figure 3. Figure 3 When the sum of three is zero, the knot is at the origin. We have A + B + C = 0 (10) 35

Forces A and B are created by the weights of two known masses. To determine the unknown force C , we can rewrite Equation (10) in the following form: C = -( A + B ) =(- A ) + (- B ) (11) Then the method described above can be directly applied to determine C . Procedure A Set up the force table on a level plane. Attached three pulleys on the rim of the round table. B . Hang the following masses on two of the pulleys and clamp the pulleys at the given angles: Force A : 20 g at 0  Force B : 10 g at 270  Place the third pulley (Force C ) at 150  with 20 g on the hanger. Add masses to this hanger until the knot is centered at the origin. Due to a small friction in the pulley, you may help the gently tap the strings to help initiate the motion. To correctly calculate the forces, the mass of the hanger (5 g) must also be included in each force. The force in Newtons equals the mass multiplied by 9.8 m/s 2 . C. Place two pulleys at 0  with 200 g on the hanger and 210  with 250 g on the hanger. By trial and error, find the angle for the third pulley and the mass, which must be suspended from it that will balance the forces. Again, tapping the strings to help Record the angle and mass. Calculations and Conclusions 1. Calculate the magnitudes and the angles of the unknown vectors for the experimental conditions of A to C . 2. Tabulate the calculated and measured results. 36

LABORATORY EXERCISE #4 Data Sheet: Vector Analysis Date:_____________ NAME:______________________________Partners:________________________ Instructor's signature:____________________ Procedure B Force A Force B Force C Experimental Force C Calculated A  a B  b C  c C  c Kg Kg Kg Kg N N N N Newton = 9.8 Kg x m/s 2 . Don’t forget to include the mass of the hangers into the calculation. Show the calculation of Force C using the space below. Procedure C 37

Force A Force B Force C Experimental Force C Calculated A  a B  b C  c C  c Kg Kg Kg Kg N N N N Show the calculation of Force C using the space below. LABORATORY EXERCISE #5 38

Tension Force and Motion Objectives: To understand the relation between the total force acting on an object and its acceleration. Equipment: Sensor cart (green) with hook, track, end-stop, 20 g and 50 g cylindrical masses with hooks, string, pulley wheel, and electronic balance. Setup: Pulley at one end of track, end-stop on the other end, hook on cart. Introduction According to Newton’s Second Law, the acceleration of an object is proportional to the total force acting on the it. In this experiment you will look at a situation in which the force is a tension force. A cart on a track will have a string attached to it, and the string will go over a pulley wheel and have a weight hanging from it. You will study the acceleration of the cart as a function of how much mass is hanging from the string. You have to be careful, though, because if the mass hanging from the string is m , then the tension in the string is not mg once the cart starts to accelerate. Before you come to this lab, you should work out the following problem: A cart of mass M moves on a frictionless track. It is attached to a string, which goes over a pulley wheel, and has a mass m hanging from it. At t=0 s the cart starts from rest and accelerates. What is the tension in the string when the cart is held stationary? What is the tension in the string when the cart is released and accelerate toward the pulley? What is the acceleration of the cart? Express the tension and acceleration using m , M and g . Procedure: 1. Open Graphical Analysis program. Select “Sensor Data Collection”. Select the cart number to Connect. Under “Sensor Channels”, enable “Force” and “X-axis acceleration” and disable other items. Then click “Done. You will see two graphs: Force- and 39

Acceleration-Time graphs. 3. Force sensor calibration Place the cart on the track with the arrow pointing toward the pulley. To improve accuracy, the force sensor needs to be calibrated. a. Click “Force” at the lower right corner. Select “Calibration”. b. Hold the cart and attach the string to the hook on the force probe. Run the string over the pulley wheel and hang a 20 g mass from it. Type 0.196 N. then click “KEEP”. c. Hang the 50 g mass on the string and type 0.49 N and click “KEEP”. Then click “APPLY”. The sensor is now calibrated. 4. To verify the calibration, hold the cart and hang the 20 g mass on the string. Click “Collect”. You will see the force reading on the graph to be 0.196 N. 5. Hold the cart with the 20 g mass on the string, click “Collect”, and wait for three seconds before releasing the cart. Catch the cart before it strikes the end stop. You will see the value of the force, which is the tension force in the string, changes when the cart is released. Use the View Statistics feature from the Graph Options at lower left of each graph to find the average value of the tension both before and after you release the cart. In addition, use View Statistics to find the average value of the acceleration after the cart is released. Do this run three times and compute the average value of your results for the tension force and the average value of your results for the acceleration. Take a screenshot of a representative screen. 6. Repeat steps 5, but with 50 g and then with 70 g (=20 g+50 g) hanging from the string. Find the average values of the tension force and the acceleration for each case. 7.Use the electronic balance to find the combined mass of the cart. Plot (by hand) your force and acceleration data. On the x axis plot the acceleration and on the y axis plot the tension force. You will have three points, one each for hanging masses of 20 g, 50 g, and 70 g. Fit the best straight line you can to your data, and find its slope. To what quantity should the slope correspond (assuming Newton’s Second Law is correct), and can you compare it to any other quantity you have measured? If you can, do so. Does your data support Newton’s Second Law? 8. If you have done the problem at the beginning of the lab, you should have a formula that gives you the tension force in terms of the combined mass of the cart and force probe and the hanging mass. Make a table comparing the calculated values for the tension and the measured values. 40

LABORATORY EXERCISE #5 Tension Force and Motion Date ______________ NAME:______________________________Partners:________________________ Instructor's Signature:____________________ Paste a screenshot of force and acceleration graphs for cart pulsed by 20 g of mass over a pulley. Tension before release=__________________ Tension after release=+__________________ Acceleration=__________________________ 41

Record of measured tension and acceleration data of cart pulled by three different masses over the pulley Tension Before Release Tension After Release Acceleration After Release 20g Ave: Ave: Ave: 50g Ave: Ave: Ave: 70g Ave: Ave: Ave: The mass of the cart =___________________ 42

Plot measured tension force vs acceleration. Choose suitable scales and units and mark the axes clearly. Fit the best straight line you can to your data, and find its slope. Slope=_____________________ To what quantity should the slope correspond (assuming Newton’s Second Law is correct), and can you compare it to any other quantity you have measured? If you can, do so. Does your data support Newton’s Second Law? If you have done the problem at the beginning of the lab, you should have a formula that gives you the tension force in terms of the combined mass of the cart and force probe and the hanging mass. What is the formula? You will be asked to show and explain how you arrived at the formula. Compare the calculated values for the tension according the formula and the measured values. Calculated Tension Measured Tension 20 g + m cart = 50 g + m cart = 70 g + m cart = 43

LABORATORY EXERCISE #6 Conservation of Energy and the value of g Objectives: To study when mechanical energy is conserved and when it is not Equipment : Sensor cart track, friction block, electronic balance, meter stick Figure 1 Experimental setup Introduction In this experiment we are going to consider two kinds of energy, kinetic energy and gravitational potential energy, and their sum, which is the total mechanical energy. We are going to apply these concepts to a cart moving down an inclined track. The cart will start from rest, and then accelerate down the track. As it does, it speeds up, and its height above the table decreases. This means that its kinetic energy increases and its potential energy decreases. We want to see if these changes compensate each other so that their sum stays the same. The second part of the experiment is the same as the first, except that a friction block will be attached to the cart. We will again want to see if the mechanical energy is constant. From class you should remember that if an object of mass m is moving with a velocity v, then its kinetic energy is given by 1 2 m v 2 . An object of mass m, which is a height h above the origin, has a gravitational potential energy given by mgh , where g=9.80 m/s 2 . If there is no other source of potential energy around (like a spring, for instance), then the total mechanical energy of the mass is just the sum of the two, E = 1 2 m v 2 + mgh If there is no friction present, this quantity will be conserved. If there is friction present, the mechanical energy will decrease, and we will have that (work done by friction)=(final mechanical energy)-(initial mechanical energy) . 44

The work done by the friction force, F f , is just – F f x , where x is the distance the object moved. Because the work done by the friction force is negative, the above equation implies that the final mechanical energy is less than the initial mechanical energy. Procedure Conservation of Energy 1. Slip the spacer block(s) under the leveling feet as shown in Figure 2. (Stack multiple blocks to form a total thickness greater than 10 cm.) Measure the distance between the leveling feet, L , the elevation of the higher end, H , and the mass of the cart , M . Calculate the angle θ . Place the cart on the track. The arrow should points to uphill. Figure 2 2. Open Graphical Analysis software. Connect the sensor to the computer. You shall see two graphs: position-time and velocity-time graphs. Click Position at the low right corner. Choose Reverse to make downhill direction positive. 3. You will be asked to hold the cart all the way up the ramp and then let it go. Before doing that, sketch your predictions for the kinetic energy versus time plot, the potential energy versus time plot, and the total mechanical energy versus time plot. 4. Now while holding the cart all the way up the ramp, click Position at the lower right corner of the screen to Zero the position, click Collect and let the cart go. Catch the cart before it hits the end stop. 5. Use the magnifiers to zoom in. Take a screenshot. 6. Record the position and velocity readings at five different times. Also record the 45

acceleration readings from the Velocity graph using Tangent under Graph Options. 7. Calculate the potential energy and kinetic energy and total energy and plot the energies as a function of time. In the calculation for potential energy, g is assumed to be 9.8 m/s/s. Compare the predicted and measured graphs and comments on any discrepancies. Measuring g 8. So far, we assume that the value of g is the accepted value of 9.8 m/s/s. From the vector diagram shown in Figure 3, the acceleration of cart is the component of g along the direction of the ramp. Thus the acceleration measured in Step 6 can lead to the value of g using the relation: g = a sinθ Figure 3 How does the value of g obtained this way compared to the accepted value? Friction 9. Now attach the friction block to the cart as shown in Figure 1. 10. Lightly press the friction block to stop the cart from sliding down hill. What is the friction when the cart is not moving? 11. You will be again asked to hold the cart all the way up the ramp and then let it go. Sketch your predictions for the kinetic energy versus time plot, the potential energy versus time plot, and the total mechanical energy versus time plot. 12. Now while holding the friction block, zero the position of the cart, click Collect, and let 46

the cart go. Use the magnifiers to zoom in. Take a screenshot. Record the position and velocity readings at five different times and calculate the potential energy and kinetic energy and total energy and plot the energies as a function of time. Use the fact that the change in the mechanical energy is equal to the work done by friction to find the value of the friction force acting on the cart. This force is due to the friction block. Use the electronic balance to find the mass of the friction block, and then find the coefficient of dynamic friction between the friction block and the track. 47

LABORATORY EXERCISE #6 Conservation of Energy Date ______________ NAME:______________________________Partners:________________________ Instructor's Signature:____________________ Distance between the leveling feet =_________________________ Elevation of the higher feet, H =__________________ Mass of the cart, M =___________________________ Predicted energy-time relation and explain why 48

Paste the screenshot of the graphs obtained in Step 3. Record position and velocity readings in five different times on the graphs and complete the following table. Table I Time Position x Height h = xsinθ Potential Energy = mgh Velocity v Kinetic Energy ¿ 1 2 m v 2 Total Energy ¿ PE + KE 1 2 3 4 5 Record the acceleration of the cart :________________________________________ 49

Plot kinetic energy, potential energy and total energy in Table I as a function of time and compare with the predicted graphs. Measuring g Based on the acceleration obtained from the Velocity graph, find the value of g. 50

Effect of friction What is the static friction when the cart on the ramp is not moving? Provide an analysis to support your answer. In the case of a cart remain stationary on the ramp, does the static friction depend on how hard you press the friction block? Draw a conclusion from the predicted and measured energies vs time graphs. 51

Paste the screenshot of the position and velocity graphs of cart motion with friction. Record position and velocity readings in five different times on the graphs of cart motion with friction and complete the following table. Table II Time Position x Height h = xsinθ Potential Energy = mgh Velocity v Kinetic Energy ¿ 1 2 m v 2 Total Energy ¿ PE + KE 1 2 3 4 5 52

Plot the measured energy-time relations with friction using Table II Use the fact that the change in the mechanical energy is equal to the work done by friction over a distance to find the value of the friction force acting on the cart. Use the electronic balance to find the mass of the friction block, and then find the coefficient of friction between the friction block and the track. 53

Laboratory Exercise #7 Collisions and Momentum Objectives: To study the relation between force and change in momentum and to determine whether momentum is conserved in inelastic collisions Equipment : Two sensor carts with bumpers, yellow cart with a bumper, electronic balance, masses, track with end stops, three 125 g masses. Introduction Instead of writing Newton’s Second Law as F=ma, we can also write it as F = ∆ p ∆t . This means that if a constant force is acting on an object for a time  t, then its momentum will change by an amount F  t, called impulse. Let us assume that the force starts acting at a time t 0 . If we plot the force as a function of time, it will be zero until t 0 , F between t 0 and t 0 +  t, and then zero again after that. The area under the force versus time plot is just F  t, which is the change in momentum. If the force is not constant, it is still true that the area under the force versus time plot will be the change of the momentum of the object. When two objects collide in the absence of external forces, Newton’s Third Law guarantees that the total momentum is unchanged. If the two colliding objects are object A and object B, then when they collide, the force exerted by A on B will have the same magnitude as the force that B exerts on A, but its direction will be opposite. This means that the change in momentum of object A will be the opposite of the change in the momentum of object B, so that the sum of their changes in momentum is zero. Therefore, the total momentum before the collision is the same as the total momentum after the collision. In terms of equations, if p A =m A v A is the initial momentum of object A p B =m B v B is the initial momentum of object B, p A =m A v A is the final momentum of object A, and p B =m B v B is the final momentum of object B, then p A +p B =P A +P B . In this lab, we want to verify Newton’s Third Law, show that the change in momentum is equal to the area under the force versus time graph, and study momentum conservation in an inelastic collision. The collision will involve two carts, one initially moving and the other sitting still, that stick together when they hit. This means that if the initially stationary cart is object B, then v B =0 and V A =V B . Procedure 1. Elastics Collision You should have two cars with built-in force sensors. The bumpers are mounted on the 54

force sensors. Place them both on the track with the arrows pointing to each other. The arrow of the green cart should point to the end stop. a. Place the green cart at the end and the yellow cart in the middle of the track. You will be asked to push the green cart, not too hard, toward the yellow cart so that that they collide. Before you take data, sketch you predictions for what the force versus time plots for each cart will look like. Note that the force is positive if it is in the same direction of the arrow. b. Open Graphical Analysis software. Click Senor Data Collection to connect the two sensors whose IDs match the ones on your carts. When both sensors are connected, click SENSOR CHANNELS, enable both Position and Force sensors. Do this for both sensors. Then click Done. Now you will see three graphs on the screen. Use View Options at the upper right corner to select two graphs. Click the vertical label of the upper graph to select Velocity G and Velocity Y and deselect all other items. (G means Green and Y means Yellow) Click the vertical labs of the lower graph to select Force G and Force Y and deselect all other items. Click Force G and Force Y on the lower right corner to zero the force sensors. Click Position Y and Force Y to reverse the direction. This is done because the arrow on the yellow cart points to the opposite direction of the green cart which is taken as the positive direction. c. Now place the yellow cart in the middle of the track, click COLLECT and gently push the green cart toward the yellow cart so that that they collide. The force sensor will measure the forces on each of the carts. Catch the cart before they strike the end stop. Use magnifiers to zoom in and then take a screenshot using your mobile phone. What does the Force-Time graph tell you about Newton’s Third Law? d. We now want to see if the total momentum is conserved before and after collision. From the graph, record the velocity of the green and yellow carts before and after collision. Next we will test whether the change in the momentum of the initially motionless (yellow) cart is equal to the applied impulse which is the area under the force versus time plot. Use the electronic balance to find the mass of the cart that was initially at rest, and then make use of your velocity-time plot to find change in momentum of this cart due to the collision. This is just the momentum after the collision minus the momentum before the collision. 55

Now use the View Integral function under Graph Options at the lower left corner of the Force-Time graph to find the area under the force curve. Compare the change in momentum to the area under the force-time graph. e. Next we will repeat the same experiment except that a 375 g is loaded on the stationary yellow cart. Before you take data, sketch you predictions for what the velocity versus time plots for each cart will look like. Then repeat Steps c and d with the yellow cart carrying 375 g (3 x 125g) masses. 2. Inelastic collision: Now rearrange the empty carts so that the two ends with Velco pads face each other. You want to have the yellow cart initially moving and the green cart initially at rest in the middle of the track. The collision will cause the two carts to stick together when they hit (again, not too hard!). You can follow the step outlined in Step 1c to obtain velocity data and thereby find the momentum before and after the collision. Now do the experiment and compare the initial and final momentum. 56

Laboratory Exercise #7 Collisions and Momentum Date ______________ NAME:______________________________Partners:________________________ Instructor's Signature:____________________ Collison between two cart of equal masses Sketch predictions for what the force versus time plots for each cart will look like during collision. Green Cart Yellow cart Paste the screenshot of the force and velocity graphs during the colllision What does your Force-Time graph tell you about Newton’s Third Law? 57

Velocity Green Cart Before collision________________. After collison=__________________. Velocity of Yellow Cart Before collision________________. After collison=__________________. Compare the total momentum of the green and yellow cart before and after collision. Present your analysis. Compare the momentum gained by the yellow cart after the collision with the area under the Force-Time graph. 58

Collison with a mass on the stationary cart Sketch predictions for what the velocity versus time plots for each cart will look like during collision when a 375g of mass is loaded on the yellow cart. Green Yellow Paste the screenshot of the force and velocity graphs for the collision with a 375 g mass on the yellow cart during the colllision Velocity of Green Cart Before collision________________. After collision=__________________. Mass of Green Cart_____________ Velocity of Yellow Cart Before collision________________. After collision=__________________. Mass of Yellow Cart______________ 59

Compare the total momentum of the green and yellow cart before and after collision. Present your analysis. Compare the momentum gained by the yellow cart after the collision with the area under the Force-Time graph. 60

Inelastic Collision Sketch a prediction for the velocity plots for the two carts before and after collision. Yellow Cart Green Cart Past the screenshot Velocity Yellow Cart Before collision________________. After collision=__________________. Velocity of Green Cart Before collision________________. After collision=__________________. Compare the total momentum before and after collision. Present your analysis. 61

LABORATORY EXERCISE # 8 Part I: Simple Harmonic Motion Objectives To study the force law for a spring (Hook's Law). To verify the equation for the period of a vibrating spring. Equipment and Supplies Spring, support for suspending spring, set of weights, stop clock, meter stick, balance. M M T= k x Mg (c) Force diagram x (a) Unstretched spring (b) Stretched spring 62

Discussion When a stretching force is applied to a spring, the elongation of the spring is found to be proportional to if the elastic limit is not exceeded. The force is a restoring force, i.e., always opposite to the displacement. This empirical law is called Hook's law: where is called the force constant, or the stiffness of the spring. The general equation of motion for the mass with acceleration a which is vertically hung from the spring as seen in the figure is within the elastic limit. Here, we completely neglect the mass of the spring which in our experiment is much smaller than the mass attached. (1) If the mass is in an equilibrium state, the equation is reduced to since . The elongation x of the spring can be measured as a function of the weight hung on it when varying the mass of the suspended weight and a plot of F vs x is a straight line, as illustrated below. Force F (N) Elongation x (m) The slope of the straight line is the stiffness of the spring. In the MKS units it is . (2) If a mass, M , hung from the end of an elastic spring is pulled down and released, it will 63

oscillate up and down. This is an example of simple harmonic motion . The acceleration must not be neglected since the mass is not in equilibrium. In this case the equation of motion is rewritten as where and the displacement at time t is where depends on the mass M and the stiffness of the spring. If the time t is replaced by t +2 , then the displacement x is unchanged. This means that the displacement has the same value after time T, . This is the period of the oscillation. A is the amplitude of the oscillation and is determined by the initial displacement of the mass. Actually, since all parts of the spring also execute simple harmonic motion, the mass of the spring needs to be included in some way. Since not all parts of the spring execute the full motion, it turns out that m' , the "effective mass" of the spring is just 1/3 of the full inertial mass, m . So, the parameter M should be the sum of the mass of the hung weight, M 0 , plus 1/3 of the mass of the spring: M = M 0 + m' = M 0 + m/3 where, M 0 is the mass of the suspended weight, m is the mass of the spring, and m' is the effective mass of the spring ( m' = m/3 ). Procedure 1. F vs. x data . Measure the elongation x of the spring as a function of the weight hung on it. Vary the mass of the suspended weight from 200 grams to 800 grams in 100 g steps. Take the position when 200 g is suspended to be zero displacement ( the origin). Do not exceed 800 g. 2. T vs. M data. Hang a mass of 200 grams from the spring and make at least three determinations with a stop clock of the time for 50 complete oscillations of the mass. Use a different initial 64

amplitude in each determination. Repeat with masses of 350, 550 and 700 grams. Calculations and Conclusions A. From the data in Procedure 1 , calculate analytically and graphically the force constant of the spring and compare the two values. The analytic calculations involve using Hook’s law for each of your data points. The graphical method involves plotting F vs. x , and measuring the slope of the best straight line through the data points (see the sketch above). B. Plot T 2 vs. M , i.e., the square of the period on the vertical axis and the mass of the suspended weight on the horizontal axis. From the graph, calculate the value of the stiffness k . C. Compare the values of k found in A and B . This means, by what percent do they differ? Now, in summary, considering both measurements, what is the precision (expressed as a percent) of your measurement of k ? D. Does the period of simple harmonic motion depend on the amplitude? How well does your data justify your answer? 65

LABORATORY EXERCISE # 8 Part I Data Sheet: Simple Harmonic Motion Date: _________ Name:______________________________ Partners:______________________ Instructor's Signature: ________________________ Mass of spring: ________________ M 0 Suspended Mass (kg) F= M 0 g Force (N) x Elongation of the spring (m) .200 Take as 0 Take as 0 .300 (.300-0.200) x 9.80 = .400 .500 .600 .700 .800 66

Hung mass: 200 g 350 g 550 g 700 g Time for 50 cycles (sec) amplitude #1 Time for 50 cycles (sec) amplitude #2 Time for 50 cycles (sec) amplitude #3 Average of three above times (sec) Period, T (sec) T 2 (sec 2 ) 67

LABORATORY EXERCISE # 8 Part II: The Simple Pendulum Objective To study how the period of a simple pendulum depends on its length. To measure the acceleration due to gravity in the laboratory room. Equipment and Supplies Metal sphere suspended by a fine string, stop clock, meter stick, vernier calipers. Discussion A simple pendulum is one where all the mass is concentrated at a small "bob". A good approximation is made by using a massive weight held with a light string from a sturdy support. If the motion is restricted to "small angles", the motion is closely simple harmonic. T , the period of the motion, depends only on the length of the string, and the acceleration of gravity. Surprisingly, the period does not depend on the value of the suspended mass: where, T is the period of oscillation. L is the distance from the support point to the center of the massive bob, and g is the local acceleration due to gravity. Procedure 1. T vs. L data. Adjust the pendulum so that L is about 100 cm. Measure L using the meter stick and vernier calipers. Determine the time for 50 oscillations with the amplitude of motion less than 10 °. 2. Repeat Procedure 1. using a length approximately 30 cm, and again with L about 60 cm. Calculations and Conclusions A. Plot your data as T 2 vs. L . The theory, given by Eq. (1), predicts that your data points should fall on a straight line of slope 2 /  g . Thus by measuring the slope of the best straight line fit, you can pull out the value of g in the lab. What is the value of g you obtain, and what is the precision you claim for your measurement ( expressed as a 68 T = 2 π √ L g ( 1 )

percent) ? B. You have just finished measuring g , the acceleration due to local gravity. Express your result as in the form wxy ±z %. In exercise #2, several weeks ago, you also measured g , using an entirely different method. For the two results summarize the following information: a) claimed precision b) agreement with accepted value Do the two results agree? The key here is the criterion you used to answer the question. Add any further relevant comments. 69

LABORATORY EXERCISE # 8 Part II Data Sheet: Simple Pendulum Date ______________ Name:______________________________ Partners:__________________________ Instructor's Signature:___________________________ Nominal length Of pendulum, L 100 cm 60 cm 30 cm Actual length, L Time for 50 oscillations (sec) Period, T (sec) T 2 (sec 2 ) 70

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LABORATORY EXERCISE # 9 The Sonometer: Vibrations of a Taut String INTRODUCTION When you pluck a taut fixed string of length L, the resultant vibrations are a superposition of many simpler "standing wave" patterns, depicted below: Note that, in general,  n = 2L / n , and f n = nf 1 , meaning that all harmonics are an integer multiple of a fundamental frequency, f 1 . Also note, that the place on a standing wave where there is no vibration is called a node , and the place where there is maximum amplitude is called an antinode. PURPOSE In this exercise you will excite and study the first few vibrational modes of the taut string. You will use a sonometer , a single-string "musical" instrument. See figure below. 72

EQUIPMENT AND SUPPLIES 1. Sonometer from Pasco Scientific Company. This instrument allows a known tension to be placed on a guitar string whose vibrating length can be varied between movable "bridges." 2. A function generator. It can "drive" an electromagnet with AC current in the lower audio range. Note: The guitar string will be attracted to the AC electromagnet twice each AC cycle, because both the North and South poles are equally good at attracting the steel string. 3. "Detector" coils capable of picking up the AC motion of a steel string. 4. A dual-trace oscilloscope to display both the driving frequency and the detected string vibrational frequency. PROCEDURE A1 . Set up the sonometer system. Set the bridges 60 cm apart. Hang a 2 kg mass from the tensioning lever. Adjust the string tensioning knob so the tensioning lever is horizontal. A2 . Position the driver coil about 5 cm from one of the bridges and the detector about 35 cm away from the same bridge. A3 . Gently pluck the string with a fingertip. The detector coil will pick up an induced voltage which you can see displayed on the oscilloscope. You can excite different waveforms by plucking in different places or with a different technique. Play with this a while. You are seeing a different mix of excited harmonics (another name for the different modes corresponding to different values of n .) 73

Note: If the oscilloscope does not display, adjust the TRIGGERING: all three levers in the upper right corner should be flipped up, and the level knob pulled out (auto mode.) Also, adjust the trace position knobs as necessary. A4 . Set the signal generator to produce a sine wave and set the gain of the oscilloscope of channel B to 5 mV/cm. A5 . With the function generator amplitude set at about 12 o'clock slowly increase the frequency of the signal to the driver coils, starting from about 5 Hz. (This part will require a gentle hand and some patience.) Listen for an increase in sound from the sonometer and/or an increase in size of the detector signal on the oscilloscope screen. Frequencies that result in maximum string vibrations are the resonant frequencies. Determine the lowest frequency at which resonance occurs. This is the first or fundamental mode (n = 1). Measure this frequency and record it on the data sheet provided. Note: Because of the effect noted in the equipment section, the resonant frequency is twice the driving frequency. Note: For the n = 1 mode, you should actually hear the sonometer hum and produce enough amplitude vibration to clearly see the central antinode and the nodes at the bridges. A6 . Continue increasing the frequency to find successive resonant frequencies for each mode - at least five or six. For each resonant mode you find, locate all the nodes, and record the distance between adjacent nodes. There are two ways to locate notes: 1. Use the "detector" provided as follows: Start with the detector as close as you can to the free bridge. Watch the oscilloscope display as you slide the detector slowly along the vibrating string. When you reach a node, the amplitude on the scope will drop to a minimum. (This method breaks down when you get too close to the "driver" and start to pick up its field directly.) 74

2. Use your fingers. Watch the oscilloscope display as you slide your fleshy fingertips along the vibrating string. Even a light touch will kill the vibrations everywhere except when you touch a node, where there is no vibration to kill, in any case. (Make sure that the detector is not accidentally left under a node, or this won't work.) A7 . From your results, determine and record the wavelength of each resonant mode you find. Make use of the fact that the wavelength is twice the distance between adjacent nodes. B1 . Repeat the experimental procedure A, for a string length of 50 cm, obtained by moving one or both of the bridges. Compute the average value, and quote the uncertainty of your measurement as a percent. 75

LABORATORY EXERCISE # 9 Data Sheet :Sonometer Date ______________ Name:______________________________ Partners:_______________________ Instructor's Signature:__________________________ 1. What mathematical relationship holds, for a given value of n, between the length L and the wavelength ?  Procedure A : String length, L = __________ n = 1 Driver frequency =__________ Actual string frequency f (twice driver frequency) = __________ Distance between nodes =__________ wavelength = 1 = ____________  n = 2 Driver frequency =__________ Actual string frequency f2 (twice driver frequency) = __________ Distance between nodes =__________ wavelength = 2 = ____________  n = 3 Driver frequency =__________ Actual string frequency f3 (twice driver frequency) = __________ Distance between nodes: __________ __________ Average distance between nodes =__________ wavelength = 3 = ____________  n = 4 Driver frequency =__________ Actual string frequency f4 (twice driver frequency) = __________ Distance between nodes: __________ __________ __________ Average distance between nodes =__________ wavelength = 4 = ____________  76

n = 5 Driver frequency =__________ Actual string frequency f5 (twice driver frequency) = __________ Distance between nodes: __________ __________ __________ ________ Average distance between nodes =__________ wavelength = 5 = ____________  n = 6 Driver frequency =__________ Actual string frequency f6 (twice driver frequency) = __________ Distance between nodes: __________ __________ __________ __________ Average Distance between nodes =__________ wavelength = 6 = ____________  n = 7 Driver frequency =__________ Actual string frequency f7 (twice driver frequency) = __________ Distance between nodes: __________ __________ __________ ________ Average distance between nodes =__________ wavelength = 7 = ____________  2a. Plot your values of the resonant frequencies versus n, the mode number. On the same graph, draw the best straight-line fit that goes through the origin. This line is the theoretical result, fn = n f1 2b. The slope of this best straight-line fit to the data (which passes through the origin) is just the fundamental frequency (f1) abstracted from all of your data, not just your first measurement. How good is the agreement? Procedure B: Use your own data sheet for this part. Take all necessary data, and present data and calculations in neat tabular form. 3. To what extent do your results in Procedure B verify the relationship given in the answer to question 1? 4. For each mode you identified and recorded in the data sheets, compute the wave speed. LABORATORY EXERCISE # 10 77

Properties of Sound Waves: Speed and Standing Waves To prepare for this lab please familiarize yourself with the following: speed of waves, standing waves and resonance. These topics can be easily located at the website “The Physics Classroom” Anyone observing a trombone player recognizes immediately that the musician changes the pitch by altering the length of the convoluted tube ( If you haven’t, check out this Youtube link https://www.youtube.com/watch?v=ZxODzxY6AvI ) In physics parlance what is being done is that the musician is adjusting the standing wave within the trombone by altering the length of the resonance chamber. This in turn alters the wavelength and frequency of the sound wave. Our ears detect the change in frequency as a change in pitch. In this lab we will be using our ears to detect the resonant frequency of an air chamber and by doing so, determine the wavelength of the sound wave. We can then easily calculate the speed of sound in air. We are also going to see how cooling the air (changing the property of the medium) affects the speed of sound. Since you are working with liquid be prepared to do immediate cleanup of surfaces after any spills. A Determine the Speed of Sound In a Tube Apparatus : 4.5 cm wide PVC tube, Plastic 2-liter graduated cylinder, Striker, 3 tuning forks 256 Hz,512 Hz. and a tuning fork of an unknown frequency, red and blue dyes, water that is at both refrigerated/ cooled and at room temperature. DO NOT STRIKE THE TUNING FORKS ON ANY SURFACE!!! Strike the tuning forks with the mallet. Your lab instructor will demonstrate. First fill the graduated cylinder to 800 ml mark. PUT 3 DROPS OF RED DYE INTO THE WATER AND GENTLY SWIRL IT AROUND . This improves visual acuity. Insert the PVC tube making sure the water level reaches just below the top. PLACE THE GRADUATED CYLINDER WITH THE TUBE INSIDE ON THE 78

CHAIR, NOT ON THE TABLE. This improves the ergonomics of the procedure. Practice lifting the tube partially out of the water and holding it for 15 seconds. Then return it into the cylinder. The set up should look something like this: Take the 256 Hz tuning fork and strike it with the striker. Practice it a few times. When you are comfortable with this, strike it one more time. While holding the fork with one hand, raise the tube out of the water with the other hand. Bring the tuning fork over the tube, holding the fork horizontally over the tube. SLOWLY raise or lower the tube. Carefully listen for a sudden change in the amplitude of the sound. Do not be afraid to put your ear as close as possible to the pipe. When the amplitude increases and decreases sharply, stop. While holding the tube at this height, your lab partner should measure the length “L” of the air chamber that is created. Length L is measured from the top of the top of the tube to the surface of the water. Play with this a bit to see how varying the length of the air chamber affects the amplitude of the sound. Practice this a number of times so that you feel comfortable locating the point of maximum amplitude. Once you feel confident 79

about locating this spot take 5 separate measures of the length and record the average the data in Table 1. Here, the air chamber has one open end at the top and a closed end ( the water) at the bottom. In this case, the wavelength is four times the length L. Reading ƛ the text will explain why this is so. Note that there is a small correction factor included in the equation to account for the magnitude of the inner diameter “d” of the tube. of the air chamber. Remember that d is in meters not centimeters. Now calculate the wavelength.  = 4 x ( L +0.3 d) Enter that value in the space provided in Table 2. Now calculate the speed of the sound wave for both the 256Hz and 512Hz conditions using the formula below. Remember to convert your length measured in centimeters to meters. V=  x f Trial 250Hz cm. 500Hz cm. Unknown cm. 1 2 3 4 5 L Avg. Table 1 256Hz 512 Hz f unknown = Hz L avg m m. ƛ V m/s V avg = Table 2 80

Now that you have made a determination of the speed of sound, see if your results are precise. Take the tuning fork with an unknown value and repeat the previous procedures for determining the wavelength of the unknown tuning fork.. Divide the wavelength value of the unknown frequency into the previously obtained average speed to obtain the frequency of the unknown tuning fork. f unknown = V avg /  Once everyone in the class has completed this procedure, your lab instructor will tell you the actual frequency of the tuning fork. Calculated frequency of the unknown tuning fork__________________________Hz. Actual frequency of the unknown tuning fork __________________________Hz. Calculate the % error using the actual value as the accepted value and your derived value as the experimental value. % Error freq =_______________________________ B Speed of Sound and Temperature The speed of a sound wave also depends on the temperature of the air. You can compare your calculated value of the speed of sound to its theoretical value based on the temperature of the room ( in Celsius) Room Temperature=______________C. ( The thermometer is located in the front of the room.) Compared to zero Celsius, this value will become T. At zero degrees Celsius in air at ordinary pressure, the speed of sound is accepted as 331 m/s. Now calculate the speed of sound at the higher room temperature “ T” use the following formula: V = 331 + (0.6m/s)T ℃ Record the average value of the speed of sound that you obtained from the two known tuning forks below and compare it the temperature dependent value. 81

V Temp = ____________m/s V Average = ____________m/s % Error_________________ If they are close (within 10%), congratulations. Otherwise, list some factors that might contribute to significant disparities between your values and theoretical values. Please don’t just say “human error”. Factors: C Changing the speed of sound by cooling the tube. Since the speed of sound is dependent on the temperature of the air, changing the temperature, should change the wavelength of the standing wave. We are going to see how much the speed of sound changes as the air chamber is cooled. For this step use only the 256 Hz tuning fork. Now get ready to cool the air chamber. Empty the cylinder of the room temperature water you have just used. Go to the refrigerator and take out one bottle of cooled water. Notice that the cold water is tinted blue with a food dye. Pour cold water from the cold water bottle to the 800 ml mark. Place the tube in the water and give it a moment to cool. Prepare to measure and record air chamber lengths as you did before. Since the tube will be colder, it will cool the air inside as well. To anticipate the warming of the air chamber as the tube is lifted, have the tuning fork and meter stick positioned for rapid measurement. Lift the tube as before and strike the tuning fork. As before, hold it horizontally over the tube as you raise it. When you detect the resonant sound, measure the length of the air chamber. Lower the tube back into the cold water. Do this for five trials and record the data. Then determine the average value of the length. Trial 1 2 3 4 5 Length cm Length Average ________________cm = _____________m 82

Calculate the wavelength using the equation: = 4 x ( L +0.3 d) ƛ Calculate the experimental speed of sound from V exp = x f ƛ Now using data from the 256 Hz tuning fork, compare the calculated speed of sound in cool air to the speed of sound obtained in warm room temperature air. V warm = _____________m/s V cool = __________________m/s Theoretically, the speed of sound changes by 0.6 meters per second per degree Celsius. While it is not feasible to actually measure the cooler temperature inside the chamber, we can get an estimate of how cold the air inside the chamber is. Using your value of V cool and the equation V cool = 331 + (0.6m/s) 𝚫 T ℃ , determine the temperature for your chamber. On the board, let the class list the temperature calculated by each table. Discard the highest and lowest values and find the average of the remaining values. Show your work. Air Chamber Temperature Class Average ________________Celsius 1. Investigate how the speed of sound is directly proportional to the temperature of the air chamber. Be brief. 2. Investigate how the speed of sound in air changes with changes in atmospheric pressure. 3. Describe two other instruments that alter their resonant frequency by adjusting the length of the air chamber. 83

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LABORATORY EXERCISE # 11 The Mechanical Equivalent of Heat Objective The purpose of this Laboratory Exercise is to prove the mechanical equivalence of heat and to verify that 1 kcal = 4186 Joules. Equipment and Supplies Commercial apparatus is provided which consists of the following: 1. An aluminum cylinder whose specific heat you know, whose mass you will soon measure, and whose temperature is conveniently sensed by a built-in thermistor. [A thermistor is a resistor whose resistance varies with temperature]. You are provided with a digital ohmmeter and a Table to tell you what temperature corresponds to what resistance. This often requires interpolation . If you are rusty on interpolation, ask your instructor. 2. A hand-crank and digital counter to record the number of revolutions. 3. A nylon rope with a "10 kg" weight at one end. The rope is wrapped a few times around the aluminum cylinder so that tangential frictional forces are set up between rope and aluminum cylinder when you turn the crank. CAUTION: Allowing the 10 kg weight to rise more than a few inches off the floor is dangerous both to student feet and to the delicate one-way digital counter. CAUTION : The Proper Procedure for turning and stopping the crank will be demonstrated by your instructor. It may look simple enough, but in fact it is easy to hurt yourself or damage the equipment. Respect this warning! Discussion Mechanical Work, which for linear motion is simply "force times distance", takes on the following expression for circular motion: W = (Torque) x angle, W = (Tangential force x Radius) x angle, W = F · R · θ , where θ is in radians. For example, a force of 2.00 N acting for one complete revolution (2 π radians) around a circle of radius 1.00 meter does an amount of work given by, W =F · R · θ = (2.00 N)(1.00 m)(2 π radians) = 4 π J = 12.6 Joules. The heat energy Q transferred to a body of mass m and heat capacity c, will raise its 86

temperature by an amount ΔT degrees, where Q = mcΔT For example, if the temperature of 0.10 kg of Aluminum (c = 0.220 kcal/kg °C) is seen to rise from 21.5 °C to 38.5 °C, we can conclude that an amount of heat Q must have been added to the aluminum, where Q = m cΔT = (0.10 kg)(0.220 kcal/kg °C)(38.5 °C - 21.5 °C) = 0.38 kcal. Procedure A. Introduction: A1 . Familiarize yourself with the apparatus. Find all the parts in the diagram below. A2 . Note how sensitive the thermistor resistance is to the touch of your hot little hands, and how quickly or slowly thermal equilibrium is approached. Now you know the timescale of the response of this equipment. A3 . When confident, unscrew the black knob that holds the aluminum cylinder in place and weigh the cylinder. Measure its diameter alone, and the diameter of the cylinder plus rope, when tightly wound. A4 . Weigh the "10 kg" mass to 3 significant figures. 87

B. "Quick and Dirty" Run B1 . Note the approximate initial temperature of the aluminum cylinder. Look-up what thermistor resistance corresponds to a "target" final temperature about 8 Celsius degrees higher. Your aim is to do enough mechanical work to heat up the aluminum cylinder to the target temperature. B2 . Reset the counter. Enter the actual initial temperature in your data sheet and start cranking vigorously! Now you are finally getting "physical" in your physics class! B3 . When you get to your target temperature, stop cranking BUT DO NOT LET GO OF THE CRANK. Allow the "10 kg" weight to drift to the floor before slowly returning the crank to its equilibrium position. Do not let go prematurely or the crank will snap back, possibly damaging the plastic clicker and invaluable body parts. Continue to watch the thermistor resistance carefully and enter in your data sheet the lowest value (highest temperature) it drifts to. B4 . Record the total number of turns and complete the calculations called for in your data sheet. C. Serious Run Even when done by an expert student, one of the major sources of error in Procedure B is that some of the heat generated will go to heat up the surroundings rather than the aluminum cylinder. To counteract this, in this procedure you will precool the cylinder so that during your run the temperature of the block will be as much below room temperature as above it. Then there should be as much heat gained from the surroundings as lost to them. It is by tricks of this nature that successful experiments are accomplished. C1 . Note the temperature of the room. Plan your target initial and final temperatures to be about 8 °C below and above room temperature, respectively. C2 . Remove the aluminum cylinder and cool it in the ice chest provided. Use plastic baggies, if available, to prevent wetting. You may not need to cool it all the way to 0 °C. In any event, the heat from your hands can now be skillfully used to make-up for too much precooling. NOTE: Carefully dry any condensed water from the aluminum cylinder with the tissue paper provided. Do this before re-wrapping the friction rope. Evaporating water could carry away a considerable amount of heat; heat that would otherwise go to warm up the aluminum. Now, re-assemble the apparatus, reset the counter and get ready. Watch the ohmmeter carefully as the thermistor resistance drifts toward your predesignated "target" for the initial temperature. When it gets there, start cranking vigorously. Crank until the temperature reaches approximately one degree short of your designated final temperature. Then crank slowly until you reach your target. Record the actual maximum temperature (minimum resistance) reached. Record the total number of turns. REMEMBER THE SAFE STOPPING PROCEDURE. A MISTAKE CAN RESULT IN YOUR HURTING YOURSELF. C3 . Compute the work done, the heat input, and the mechanical equivalent of heat as guided by your data sheet. Compute the percent deviation from the accepted value. 88

D . Consider the possible reasons (mechanisms) why your answer is different from the accepted value. Do you have any suggestions to improve this experiment? Resistance versus Temperature for your thermistor: Res. T Res. T Res. T Res. T (Ω) (°C) (Ω ) (°C) (Ω ) (°C) (Ω ) (°C) 351,020 0 95,447 26 30,976 52 11,625 78 332,640 1 91,126 27 29,756 53 11,223 79 315,320 2 87,022 28 28,590 54 10,837 80 298,990 3 83,124 29 27,475 55 10,467 81 283,600 4 79,422 30 26,409 56 10,110 82 269,080 5 75,903 3 25,390 57 9,767.2 83 255,380 6 72,560 32 24,415 58 9,437.7 84 242,460 7 69,380 33 23,483 59 9,120.8 85 230,260 8 66,356 34 22,590 60 8,816.0 86 218,730 9 63,480 35 21,736 61 8,522.7 87 207,850 10 60,743 36 20,919 62 8,240.6 88 197,560 11 58,138 37 20,136 63 7,969.1 89 187,840 12 55,658 38 19,386 64 7,707.7 90 178,650 13 53,297 39 18,668 65 7,456.2 91 169,950 14 51,048 40 17,980 66 7,214.0 92 161,730 15 48,905 41 17,321 67 6,980.6 93 153,950 16 46,863 42 16,689 68 6,755.9 94 146,580 17 44,917 43 16,083 69 6,539.4 95 139,610 18 43,062 44 15,502 70 6,330.8 96 133,000 19 41,292 45 14,945 71 6,129.8 97 126,740 20 39,605 46 14,410 72 5,936.1 98 120,810 21 37,995 47 13,897 73 5,749.3 99 115,190 22 36,458 48 13,405 74 109,850 23 34,991 49 12,932 75 104,800 24 33,591 50 12,479 76 100,000 25 32,253 51 12,043 77 89

LABORATORY EXERCISE # 11 DATA SHEET : Mechanical Equiv. of Heat Date:_____________ Name:_________________________________ Partners :_____________________ Instructor's Signature:__________________ A. Fixed parameters Specific heat of Aluminum, c = 0.220 kcal/kg °C Mass of Aluminum cylinder, m = ___ Diameter of cylinder, D 1 =_________ Diameter of cylinder plus rope, D 2 =_______ Average, Do = ( D 1 + D 2 )/2=_________ Average Radius, R = Do/2=_________ Mass of "10 kg" mass, to 3 significant figures, m=__________kg Weight of "10 kg" mass, to 3 significant figures, F=__________Newtons B. "Quick and Dirty" Run B1 . Approximate initial temperature: Thermistor Resistance =_________ Corresponding Temperature = __________ "Target" final temperature: Target Temperature =_________ Thermistor Resistance=__________ B2 . Actual starting temperature: Thermistor Resistance =__________ Corresponding Temperature =________ B3 . Actual final temp.: Thermistor Resistance =__________ Corresponding Temperature =___________ B4 . Number of turns, N =__________ Total angle, θ = 2 N=___________radians  B5 . Computed total work you did, W = F R θ =___________Joules Computed heat input to Aluminum cylinder, Q = mcΔT = ___________kcal Your computed Mechanical Equivalent of Heat, (ratio of above two results):_______________ 90

What is the percent deviation of your result from the accepted value of 4186 J/kcal? ____________ % C1 . Room Temperature today____________ °C Target initial temperature____________ °C Corresponding Thermistor Resistance__________Ω Target final temperature_____________ °C Corresponding Thermistor Resistance____________Ω C2 . Actual initial temp.: Thermistor Resistance =__________ Corresponding Temperature = ___________ °C Actual final temp: Thermistor Resistance =__________ Corresponding Temperature = ___________ °C Number of Turns__________Total angle, θ = 2 N =____________radians  C3 . Compute the total work you did against friction: W = FRθ = ___________Joules Compute the amount of heat that must have been transferred to the Aluminum, Q = mcΔT = ____________kcal Compute the Mechanical Equivalent of Heat and the percent deviation: D . Did you do better in Part C than in Part B? If not, what went wrong? Give reasons. 91

LABORATORY EXERCISE # 12 Boyle's Law Objectives To study how changing the pressure on a gas makes the volume change. To compare your results with those obtained by Bobby Boyle. Equipment and Supplies A sample of air trapped at the bottom of a glass tube by a short column of mercury. A ruler to measure the length of the air column. A barometer somewhere in the room. Discussion According to Robert Boyle, if the temperature of a fixed amount of gas is not allowed to change, the following relationship will hold as the volume, V, and the pressure, P, are changed: P · V = a constant number (1) The value of the constant depends on the number of moles of gas, the temperature, and the particular units used to measure Pressure and Volume. Boyle's Law accurately describes the constancy of the pressure-volume product for most common gases at moderate temperatures and pressures. Deviations from this Law may become significant if the pressure is too high or the temperature is too low. These limiting values vary drastically with different gases. For example, hydrogen obeys Boyle's Law at -200 °C and 100 atmospheres; sulfur dioxide does not at 20 °C and 1 atmosphere. The apparatus for this experiment is elegant in its simplicity. The short mercury column in the glass capillary serves two purposes. First, it serves as a leak-proof piston to isolate your gas sample. Second, it provides additional pressure on the gas sample as the glass tube is inclined. By varying the vertical orientation of the glass tube, you can vary the pressure of the gas sample. To see how this works, note that the absolute pressure of the enclosed air in Figure (1a) is, P = P atm + ρ g (Y 1 - Y 2 ) (2) where P atm = atmospheric pressure, ρ = the density of mercury, 13,000 kg/m 3 , and Y 1 and Y 2 = the vertical heights of the ends of the mercury column. If atmospheric pressure is expressed in centimeters of mercury, call it B cm of Hg, (obtained directly from the mercury barometer at the end of your lab room) then the 92

absolute pressure of the gas sample of Fig (1a) can be simply written as, P = ρ g (B + Y 1 - Y 2 ) (3) Life is simplest if you express the pressure of the gas in units of cm of Hg instead of N/m 2 . Equation (3) becomes, P = B + Y 1 - Y 2 (4) Now, for one more simplification: Because the cross-sectional area of the capillary is a constant, the volume of the gas is directly proportional to its length, L . As a result, Boyle's Law for your gas in the capillary is simply, ( B + Y 1 - Y 2 ) L = a constant number (5) Note: When the open end of the capillary is lower than the closed end, ( Y 1 - Y 2 ) is negative. See Figure (1b). As Equation (4) predicts, in that case the pressure of the gas will be less than atmospheric pressure. So, the extremes of pressure you can obtain are B + M and B - M. Procedure 93

1. Measure atmospheric pressure in cm of Hg using the classroom barometer. Locate the barometer, examine its parts. Adjust the barometer so that the surface of the mercury in the cup at the bottom of the instrument just touches the apex of the ivory cone. This marks the zero point of the barometer scale. Next, align the lower surface of the vernier slide with the top of the mercury column. (Use the top of the meniscus). Record the mercury level in your Data Sheet. Some days the pressure can change measurably during the course of the experiment; take another reading at the end of the lab period and use the average value if you notice a change. 2. Measure the length of the air column as you vary the pressure of the gas. For ten positions of the apparatus, chosen to yield approximately equal increments of pressure, record the length, L , of the air column, and the heights Y 1 and Y 2 . Two of the positions must correspond to the extremes of pressure. Note: The best way to measure L is to measure the positions of the endpoints of the air column, L = X 1 - X 2 . Calculations and Conclusions 1. Plot P vs. L and P · L vs. L on a single sheet of graph paper. Calculate the product P · L for each value of L you chose, then plot the graphs requested. Suppress the origin of the abscissa so that the range of L observations covers the horizontal scale. Do not suppress the origin of the ordinates, otherwise the experimental errors will be magnified on the graph. Draw a smooth curve through the P vs. L points. Draw the best straight line through the P · L vs. L points. Which graph is better suited to judge how well your data follows Boyle's Law? 2. Plot P vs. 1/L. Draw the best straight-line fit through the origin, representing "the theory", i.e., Boyle's Law. How well does your data follow the accepted theory? Comment on possible reasons for disagreement, if there is any in your case. 94

LABORATORY EXERCISE # 12 Boyle's Law: Data Date ______________ NAME:______________________________Partners:________________________ Instructor's Signature:____________________ Boyle's Law B (Air Pressure, cm of H) Temperature °C Trail #! Trail #2 Average Length of mercury column, M=______________________________ Trial # X 1 cm X 2 cm L=  x cm Y 1 cm Y 2 cm  Y= Y 1 -Y 2 P, cm =B+  Y P x L cm 2 LABORATORY EXERCISE # 13 95

Part I: Heat of Fusion of Ice Objective To determine the heat of fusion of ice by the method of mixtures. Equipment and Supplies Calorimeter, balance, thermometer, ice cubes. Discussion The change of a substance from the solid to the liquid state, called melting or fusion takes place at a definite temperature characteristic of each substance. During the transition no change in temperature occurs. To bring about this change of state a definite amount of heat must be supplied from the surroundings for each unit mass melted. This quantity of heat is called the (latent) heat of fusion of the substance. Thus the amount of heat required to melt a solid of mass m and heat of fusion H f without changing its temperature is mH f . This is also the amount of heat given up to the surroundings when the liquid changes to the solid state. In the metric system H f is measured in calories/gram or kilocalories/kilogram. (Numerically these are the same. ) If two substances at different temperatures are mixed in a container thermally insulated from its surroundings, the final temperature of the mixture will be between the original temperatures. From the principle of the conservation of energy, the heat lost by the warmer substance must equal the heat gained by the colder: Heat gained by colder body = Heat lost by warmer body To determine the heat of fusion of water by this method of mixtures, a piece of ice at 0 ° C is dropped into a calorimeter containing warm water. The initial temperature of the water, its final temperature after being cooled by the ice and the masses of the ice, warm water and calorimeter are measured. If the specific heat of the calorimeter is known, the heat of fusion of water may be determined using the fact that the heat absorbed by the ice in melting, plus the heat needed to raise the temperature of the resulting cold water to the final temperature of the mixture equals the heat lost by the warm water and calorimeter: [ m H f + m c ΔT ] ICE = [ m' c ΔT ''] WARM WATER + [ m" c" ΔT''] CALORIMETER Here, m , m' and m'' are the masses of the ice cube, the warm water and the calorimeter, respectively. Also, c and c'' are the specific heats of water and aluminum. Finally, ΔT is the temperature rise of the ice, while ΔT '' is the temperature fall of the warm water. 96

Procedure 1. Weigh the inner calorimeter vessel and stirrer, fill it approximately two-thirds full of water about five degrees above room temperature and weigh again to determine the mass of the water. Place the calorimeter vessel inside the calorimeter and insert a thermometer to determine the temperature of the water and vessel. 2. Stir the water in the calorimeter well and record its temperature. Dry an ice cube, place it in the water, replace the calorimeter cover and, when the ice has melted, stir the water and record its final temperature. 3. Weigh the calorimeter vessel, stirrer and contents to determine the mass of the ice used. Calculations and Questions A. Calculate from your data the heat of fusion of water. B. Compare the value found in A and the generally accepted value for the heat of fusion of water. C. Why was it desirable to have the initial temperature of the water slightly above the temperature of the room? Note: The data sheet for this experiment and the next are combined into one. You will find it following the next experiment. 97

LABORATORY EXERCISE # 13 Part II: The Specific Heat of a Metal Objective Determine the specific heat of a metal by the method of mixtures. Equipment and Supplies Calorimeter, steam boiler, metal shot, balance, two thermometers, small beaker. Discussion The "specific heat", c , of a substance is the amount of heat required to raise the temperature of a unit mass one degree. If the specific heat is independent of the temperature ** , the amount of heat, Q , required to raise the temperature of a substance whose mass is m and whose specific heat is c , from an initial temperature T i to a final temperature T f , is, Q = m c ( T f - T i ) = m c ΔT Since the calorie (kilocalorie) is defined as the amount of heat required to raise the temperature of 1 gram (1 kilogram) of water 1 ° Celsius from 14. 5 ° to 15. 5 °, the specific heat of water is 1 cal/gm °C (1 kilocal/kgm °C). If two substances at different temperatures are mixed in a container thermally insulated from its surroundings, the final temperature of the mixture will be between the original temperatures. From the principle of the conservation of energy, the heat lost by the warmer substance must equal the heat gained by the colder: Heat gained by colder body = Heat lost by warmer body. You will determine the specific heat of a metal by this method of mixtures. A mass of metal pellets at a high temperature is dropped into a calorimeter containing cold water. You will measure the initial temperature of the metal pellets and of the water, the final temperature of the mixture and the masses of the components (metal pellets, water and calorimeter.) If the specific heat of the calorimeter is known or if the calorimeter is made of the metal whose specific heat is being determined, the specific heat of the metal may then be determined by using the fact that the heat lost by the metal must equal the heat gained by the water and the calorimeter. Procedure * ** Specific heats actually vary slightly with temperature. However, for the range of temperatures encountered in this experiment and for the expected precision, this variation may be safely neglected. 98

1. Heat the water in the boiler. Weigh the boiler cup, fill it approximately two-thirds full of metal pellets (copper or aluminum) and weigh again to determine the mass of the pellets. Place the cup in the boiler and insert a thermometer in contact with the pellets. 2. While the metal pellets are heating, weigh the inner calorimeter vessel and stirrer, fill it approximately two-thirds full of water which is a few degrees below room temperature and weigh again to determine the mass of the water. Place the calorimeter vessel inside the calorimeter and insert a thermometer to determine the temperature of the water and vessel. 3. When the temperature of the metal pellets has become stationary at nearly 100 °C, record its temperature and that of the water in the calorimeter. Be sure that the water has been stirred well. Then quickly pour the shot into the water being careful that no water is splashed out. Replace the calorimeter cover, stir well and record the highest temperature to which the water rises. 4. If the calorimeter is not made of the same metal as the shot, obtain from your instructor the specific heat of the calorimeter. (The calorimeter is made of aluminum.) Calculations and Conclusions A. Calculate from your data the specific heat of the metal used. B. Compare the value found in A with the generally accepted value for the specific heat of the metal used. C. Why was it desirable to have the initial temperature of the water slightly below the temperature of the room? 99

LABORATORY EXERCISE # 13 Data Sheet for both calorimetry Labs Date ______________ Name:______________________________Partners:_____________________________ Instrutor's Signature:_________________ Calorimetry (1): Heat of Fusion of Water 1. (a) Mass of calorimeter & stirrer:_________________________________gm (b) Mass of calorimeter and stirrer 2/3 full of water:__________________gm {water is ~5 ° above room temperature} (c) Mass of water (difference of above two masses): _________________gm 2. (a) Initial temperature of water, calorimeter & stirrer:__________________ °C {Here, you added towel-dried ice cubes} (b) Final temperature of water, calorimeter & stirrer and melted ice:______ °C (c) ΔT , temperature change, from the above two temperatures:_________ °C 3. (a) Final mass of calorimeter & stirrer, original water plus melted ice:_____ °C (b) Mass of ice cubes {line 3(a) minus line 1(b)}:_____________________gm Do not empty or add more water! All will be used "as is" for the next exercise ( # 14). Calorimetry (2): The Specific Heat of a Metal 100

4. (a) Initial mass of boiler cup:____________________________________gm (b) Mass of cup plus metal pellets:_______________________________gm (c) Mass of metal pellets {line 1(a) minus line 1(b)}:__________________gm 5. (a) Mass of calorimeter & stirrer {same as line 1(a) from other side}:_____gm (b) Mass of calorimeter & stirrer and cool water:_____________________gm {same as line 3(a) if no water was added or removed} (c) Mass of water { line 5(b) minus line 5(a)}:_______________________gm 6. (a) Temperature of hot metal pellets:______________________________ °C (b) Initial temperature of water and calorimeter & stirrer:_______________ °C (c) Final highest temperature of mixture:___________________________ °C 7. (a) Specific heat of calorimeter & stirrer:______________________cal/gm °C {given by instructor, if needed} 101