Lab-2_Motion on Incline_LAB

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Apr 3, 2024

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1-D Motion on Incline 1-D Motion on Incline–Constant Acceleration OBJECTIVE The objective of this lab experiment is to make simultaneous experimental observations of the position, velocity, and acceleration of a uniformly accelerating object on an inclined plane. From these data, it will be possible to study the relationship between these three quantities, and make a measurement of the acceleration due to gravity. THEORY When an object is placed on a "frictionless" inclined plane, there are two forces acting on it : the force, mg , due to gravity and the normal force , N , exerted on it by the plane. Since the plane is assumed to be frictionless, there is no component of this contact force parallel to the plane. If the x- and y-axes of the system are chosen to be parallel and perpendicular to the incline of the plane respectively, then the weight, mg , can be resolved into its components as shown in Figure 1. Figure 1 Newton's second law of motion states that: Or where , since there is no motion perpendicular to the plane: ∑ F = m × a (1) ∑ F x = m × a x ∑ F y = m × a y a y = 0 F y = N − mg × cos θ = 0

1-D Motion on Incline Page 2 Then: Thus it is the component of the weight parallel to the incline that produces the acceleration down the plane. Since the plane in Figure 1 has length, L, and is inclined to a height, h , the value of sin θ is h/L . Substituting this into equation (2) yields an acceleration down the plane of: The constant acceleration of the object on the incline can be determined with the equation of motion If the object is released from a starting point, X (0) = 0 with V x (0) = 0 , then the acceleration of the object as it moves through a distance, X , in the time, t , is given by: Combining equations (3) and (4), an equation is obtained which can be used to determine the value of g with data acquired in this experiment: Thus g can be determined by measuring the quantities X , L , h , and t . F x = mg × sin θ = ma x a x = g × sin θ (2) a x = gh L (3) X ( t ) = X (0) + V x (0) + 1 2 a x t 2 a x = 2 X t 2 (4) g = 2 XL ht 2 Updated: Summer 2020

1-D Motion on Incline Page 3 1-D Motion on Incline– Lab Report Section: _011____________ Name: ___Emily Espinoza_________________ GOAL: (briefly state what experiment(s) will be performed and with what purpose) Equipment You need the following items: 1. Ramp, 2. Meter Stick. 3. A small Ball or Empty tin can, 4. Electronic Stopwatch Figure 2. • For the ramp you can use any flat, level object such as a board or a cardboard box. It should be about 1.25 meters (4.0 ft.) long. It doesn't matter what the ramp is made of as long as it is flat and strong enough to support your tin can without bending (Be creative). • To raise one end of the ramp, use a pile of books. You may not be able to get the top of the ramp at exactly the height given in the instructions. Get it as close as you can and record the actual height in data tables A and B. Updated: Summer 2020

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1-D Motion on Incline Page 4 • It is best to use a "starting gate" to avoid imparting to the can any uphill or downhill motion. The starting gate is a pencil or small ruler which holds the can in place, to be lifted when time starts. Think of how the gates at a parking garage operate. All of this is because we want to assure that the initial speed is zero (stationary). • A countdown of "5, 4, 3, 2, 1" is a good way to begin the timing whether there is one person or two involved in the experiment. • To stop the timing, it is best to use a flat object such as a ruler or the cover of a book as a physical stop. This allows you to use your sense of hearing along with sight to coordinate the stopwatch with the stopping point. In this experiment there are FOUR VARIABLES. 1. The distance along the ramp. 2. The steepness of the incline which is measured by the ratio of height to length of the ramp. 3. The height from which the object is released on the ramp. 4. The time required for the ball to roll a certain distance down the ramp. HYPOTHESIS A: Distance is directly proportional to the square of time if acceleration is uniform. In this part the distance d down the ramp is the variable while the angle θ of slope is constant. 1. Set up the ramp with h= 0.10 m above the ground, (as shown in Figure 2.) 2. Starting with the cylinder at rest, use the stopwatch to measure the time to roll distance d = 1.0 meter down the ramp. 3. Take 6 time measurements, record in data Table A . Cross out the highest and lowest times and determine the average of the remaining four times. (Sum four times and divide by four to find the average.) 4. Repeat steps 2 and 3 for distances of 0.80 m , 0.60 , 0.40 m (see data Table A). HYPOTHESIS B: Rate of acceleration is proportional to incline of ramp. In this part the distance rolled down the ramp d and the angle θ of slope are both variables. 1. The steepness of the incline can be measured by the ratio of height to length. Measure the total length of the ramp and record in data Table B as L. 2. Copy the data from the first set of measurements in hypothesis A into the data table for hypothesis B, unless you prefer to collect that data a second time. (Time to roll 1.0 m when h = 0.1 m) 3. Raise the top of the ramp to 0.15 m . Updated: Summer 2020

1-D Motion on Incline Page 5 4. Starting with the cylinder at rest, use the stopwatch to measure the time to roll 1.0 meter down the ramp. 5. Take 6 time measurements, record in data. 6. Repeat steps 4 and 5 for heights of 0.20 m, 0.25 m. HYPOTHESIS C: Objects will reach the same speed from a given height regardless of incline. In this part the starting height is held constant while the distance rolled down the ramp and the angle of slope are varied. 1. With the top of the ramp at h = 0.25 m as in part B, find the point P on the ramp that is y = 0.10 m above the ground (see Fig 2). 2. Measure the length of the ramp from that point P to the BOTTOM of the ramp and record as d in data table (see Fig 2.) 3. Measure the time for the cylinder to roll from point P to the bottom of the ramp (start from rest as before). 4. Repeat steps 2 and 3 for ramp heights of h = 0.20 m, 0.15 m, and 0.10 m . * note that the distance d, the incline of the ramp, and point p will be different for each of the 4 trials while the height y remains constant. DATA TABLES A. Acceleration vs. Distance Calculate the average time for six trials. Discard the highest and lowest in each trial so you will average the middle four values. Enter the average time in the data table. In each case use t(avg) for the times to calculate t squared. Draw a graph of distance versus average time squared for your data. Table A B. Acceleration vs. Slope d (m) t 1 t 2 t 3 t 4 t 5 t 6 t (avg) t 2 1.00 3.23 2.73 2.77 2.42 2.85 3.16 2.88 8.29 0.80 2 2.03 1.88 1.91 1.82 2 1.95 3.8 0.60 1.6 1.51 1.83 1.62 1.52 1.65 1.6 2.56 0.40 1.29 1.31 1.19 1.41 1.32 1.33 1.33 1.77 Updated: Summer 2020

1-D Motion on Incline Page 6 L is the total length of the board, to be recorded below. Distance rolled (d) is 1.0 meter. "t 2 " means ‘t raised to the power of 2’ Height divided by length (h/L) is a measure of steepness of slope. Distance divided by time squared (d/t 2 ) is proportional to acceleration. The relationship will be linear if and only if acceleration is directly proportional to slope. Table B From data table B above, calculate the average time, the square of the average time, ratio d/t avg 2 and the ratio h/L. For each of the four heights divide the two ratios from table B (divide d/t 2 by h/L) and enter the results in the table B1 below. Table B1 Examine the ratios and decide whether or not they are constant. Then plot a graph of ( d/t 2 ) vs. ( h/L ) with h/L on the horizontal axis. L = _0.10____m d = 1 m h (m) t 1 t 2 t 3 t 4 t 5 t 6 t avg t avg 2 (d/t 2 ) (h/L) 0.10 1.72 1.74 1.79 1.77 1.75 1.74 1.75 3.063 0.327 0.10 0.15 1.44 1.41 1.43 1.45 1.43 1.42 1.42 0.49 0.49 0.15 0.20 1.25 1.28 1.24 1.23 1.24 1.22 1.24 0.65 0.65 0.20 0.25 1.11 1.09 0.97 1.14 1.10 1.11 1.02 0.82 0.82 0.25 h (m) (d/t 2 )/(h/L) 0.10 3.27 0.15 3.2733 0.20 3.25 0.25 3.288 Updated: Summer 2020

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1-D Motion on Incline Page 7 C. Speed vs. Height " y " is the constant height from which the object is rolled. " h " is the measurement to the top of the ramp as before, " d " is the distance from point P along the ramp to the bottom. See figure 2. Be sure you understand the meaning of the variables before you begin . You want the object to be rolled from the same height above the ground each time. The point of release and the distance rolled down the ramp will be different for each trial. Table C For each height, calculate the average speed at the bottom of the ramp, d/t and enter the results in the data table C1 below. If the speed is the "same" then the ratios d/t should be equal. Table C1 Comparing the ratios of d/t decide whether or not you think the speeds are the same, then draw a graph of d vs. t for each of the four slope angles. QUESTIONS QUESTIONS FOR HYPOTHESIS A . y = 0.10 m h (m) d (m) t 1 t 2 t 3 t 4 t 5 t 6 t avg 0.10 1.10 1.28 1.27 1.29 1.25 1.26 1.24 1.265 0.15 1.15 1.39 1.35 1.33 1.36 1.37 1.39 1.365 0.20 1.20 1.52 1.50 1.55 1.57 1.54 1.56 1.54 0.25 1.25 1.65 1.62 1.60 1.66 1.63 1.64 1.63 h (m) (d/t) 0.10 0.869 0.15 0.842 0.20 0.779 0.25 0.767 Updated: Summer 2020

1-D Motion on Incline Page 8 Using your data from part A, plot a graph of d (vertical axis) vs. t 2 (horizontal axis). Draw a "best fit" straight line through the points. 1.Is the graph linear? No it is not linear 2.What does it mean if the graph is linear? If the graph is linear then the object had a constant acceleration. x and y would have a relationship that is proportional 3. What does a linear graph indicate about the acceleration of rolling objects? A linear graph demonstrates the object that is rolling is at a constant rate. 4. Does your data support hypothesis A? Briefly justify your answer. Our data did not support hypothesis A. Distance and time do not have a proportional relationship. QUESTIONS FOR HYPOTHESIS B Using your data from part B calculate d/t 2 and h/L . Then plot a graph of d/t 2 vs. h/L with h/L on the horizontal axis. Draw a "best fit" straight line through the points. 1.Is the graph linear? Yes this graph is linear 2.What does it mean if the graph is linear? It means that as we increase h, the ratio d/t avg ^2 will also increase. 3.Does your data support hypothesis B? Briefly justify your answer - The data backs up the hypothesis for low h values. There is a deviation from the Linear connection as a the value of h increase. QUESTIONS FOR HYPOTHESIS C The average speed of an object under constant acceleration is distance divided by time. Plot a graph of d vs. t with t on the horizontal axis. Draw a "best fit" straight line through the points. Updated: Summer 2020

1-D Motion on Incline Page 9 1. Is the graph linear? Yes, the graph is linear 2. What does it mean if the graph is linear? If the graph is linear it indicates that the object was moving at a constant speed. As shown in the distance vs time graph it shows how far the object that was being used traveled in each of the time trials. 3. Does your data support hypothesis C? Hypothesis C was <objects will reach the same speed from a given height regardless of incline.=My data does not totally support this hypothesis since the average speeds that I got were 0.869m/s, 0.842 m/s, 0.779 m/s, and 0.767 m/s. Although the speeds are around similar averages, they are not directly proportional. GENERAL QUESTIONS 1. What are the variables in this experiment? The length of the ramp, the incline's steepness, the height at which the object was thrown, and the amount of time the ball took to roll were the variables in this experiment. 2. How did you control the can to be sure it did not roll off the side of the board? Would this have any effect on the precision or accuracy of your measurements? By adding items to the sides, we were able to regulate the object and keep it from rolling off the side by allowing it to roll in a specific direction and down straight. We believe that the precision or accuracy of our measurements would be impacted if there was nothing to the sides from where the object was being rolled. Because it wouldn't be traveling straight, the thing would roll everywhere. 3. How can you tell whether or not the points on the graph represent a linear relationship? Would you expect them to be perfectly linear? Why or why not? The graph's slope will change if any of the points are not precisely linear, as we would expect. Updated: Summer 2020

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1-D Motion on Incline Page 10 4. Briefly discuss the problems encountered in making kinematic conclusions from experimental data. Don't confuse the process of collecting data with the process of drawing conclusions from it. They cannot assume that by utilizing the constant acceleration equation, they can calculate any necessary variable, including speed or acceleration. We are unable to determine actual velocity or acceleration using kinematics' three equations of motion since acceleration is never constant in real- world situations. Note: Use Excel to plot all the graphs then copy and paste them to your report. Once you have completed the assignment, save your report as a PDF and upload using the blackboard submission on the lab course page (found under “Lab Online Submissions”) Updated: Summer 2020

Lab-2_Motion on Incline_LAB

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