Lab 11 Mass-Spring System CPI F23 (1)

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Dec 6, 2023

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11 Mass-Spring System Purpose To experimentally determine the spring constant, k , for a spring under going simple har- monic motion. Equipment • Lab Pole Clamp (1) • Long Lab Pole (1) • Right Angle Clamp (1) • Short Lab Pole (1) • Assorted Springs (5) • Mass Hanger with Assorted Masses (1 set) • PASCO Motion Sensor (1) • Computer with CAPSTONE Software (1) • Notecard (1) • Masking Tape (1 - roll) 11.1 Overview of the Lab Activity In this activity you will analyze the simple harmonic motion of various mass-spring systems and use the relationship between the period and the hanging mass to determine the spring constant, k . Imagine a spring is extended or compressed so that its end is at a position x . If the equilibrium position of the end of the spring is x o , the spring exerts a reactionary force given by Hooke’s Law: F spring = - k ( x - x o ) (32) The minus sign indicates that the spring’s force vector points in the opposite direction of the displacement. The spring constant, k (in units of N m ), represents the amount of force needed, theoretically, to stretch the spring one meter (or the ”sti ↵ ness” of the spring). In this activity, you will use a vertical system where only conservative forces act on the system. Hang one of the springs from the support rod. Place a mass hanger and a small amount of mass on the end of the spring hanging from the support so that the springs hangs with very small separation between all the loops in the spring. This is the equilibrium point, x o . Pull down on the mass gently to give the mass a small amount of displacement and let go. What do you observe? 92

When the spring is released, this restoring force accelerates the mass back toward the equi- librium position. Because the mass has momentum, it continues through its equilibrium position and extends or compresses the spring through the same amount of displacement in the opposite direction and keeps repeating this. The spring and mass are now in simple harmonic motion. To make our lives much easier, we will set up our measurement system so that the equilibrium position, x o = 0 and measure the displacement of the object to be some position from the equilibrium. Sketch a picture of the mass as it starts to move from the maximum displacement through the equilibrium and back again to the maximum displacement on the other side of the equilibrium. At what position(s) in the motion is the mass moving with the greatest speed ? At what position(s) in the motion is the mass moving with the least speed ? 93

11.1.1 Simple Harmonic Motion (SHM) When you set the hanging mass into motion, it travels in the vertical direction from max- imum displacement through the equilibrium to maximum displacement and back again, over and over. Assuming no loss in energy, the system will continue to do this over and over. One complete cycle of the motion is called the Period, T , and requires a set amount of time based on the size of the hanging mass and the strength of the spring, called k and is defined as the spring constant. If we were to attach a pen to the oscillating mass and place a piece of paper behind the mass, and then drag the paper as the hanging mass oscillates up and down you would get the following figure. Figure 1. Oscillating Mass and Wave Form The motion of the hanging mass oscillating up and down creates a wave form and is the same as an object traveling in a uniform circle. Figure 2. Period ( T ) and Angular Frequency ( ! ) From Figure 2 we can see that the period of motion is the time it takes for the hanging mass to travel one complete up - down - cycle. The frequency, f, is how many cycles can be completed each second. Frequency is the inverse of the period and is given by: f = 1 T . We have seen period and frequency before, when discussing uniform circular motion. With uniform circular motion, we used the angular frequency, ! - how many revolutions per second, to describe the motion. 94

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Simple harmonic motion is represented graphically in a very similar fashion to uniform circular motion, so it makes sense to use ! again, instead of f . The connection between the two is ! = 2 ⇡ f. This means that the connection between ! and the T is ! = 2 ⇡ T . (33) Change the mass on the hanger (approximately twice as much) and set the mass into motion using the same displacement you used before. What do you observe about the period of the motion? You should have observed that the mass on the hanger takes a longer time to make one cycle. So the size of the mass has an a ↵ ect on the oscillation. If you were to change the spring to one that was sti ↵ er, then you would also notice a change in the period of oscillation. The change in the period, T , of oscillation means that the angular frequency, ! , changes. The dependent relationship between the angular frequency, the spring force and the hanging mass is given by: ! = r k m (34) Examine Equation 34, write an expression for the angular frwquency, ! , in terms of the spring constant, k , and the hanging mass, m that does not involve the squareroot function. Examine this equation and determine the dependent and independent variables: Independent variable: Dependent variable: 95

Sketch a graph of this equation (be sure to title it and label both axes) and describe the shape of this graph: All of our assumptions and discussion so far have been for an ideal spring, one with a negligible amount of mass, that could be ignored compared to the hanging mass. One- third of the mass of the spring needs to be added to the amount of the oscillating mass when we are considering a vertically suspended system (Note: the justification for this step requires calculus beyond what is required for this course). Using the digital scales at the front of the room, measure the mass of each of the springs and record in Data Table 1. Be sure to add in the spring mass to the hanging mass when you enter data in Excel. m total = m + m hanger + 1 3 m spring 11.2 Data Collection 1. Log on to Blackboard and download and open the Lab 11 CAPSTONE file for this activity. 2. Ensure the motion sensor is turned on and connect it via Bluetooth to the CAP- STONE software. 3. Select a spring and record the color in Data Table 1. 4. Measure the mass of the spring with the digital scale, and record in Data Table 1 5. Hang the spring from the support lab rod. 6. Hang a mass hanger from the end of the spring. 7. Add a small amount of mass to the hanger so that the spring is slightly displaced uniformly in length. This is the equilibrium position, x o . 8. Add some more mass to the mass hanger. Record the total amount of mass in Data Table 1. 96

9. Position the motion sensor below the mass hanger. (Attach a note card to the bottom of the hanger if the motion sensor is returning a bad signal.) 10. Click ”Start” on the CAPSTONE software. 11. Displace the mass on the hanger by some small distance x by pulling down on the mass hanger and release. 12. Allow the mass to go through about 5-6 complete oscillations and click ”stop”. This is an example of what you should see for your data collection. 13. Select the Coordinate Tool and position the tool at the peak of one complete oscillation. 14. Then right-click the tool and select Displacement . Then drag the Displacement tool to the next peak and record the Period (the time between peaks) in Data Table 1. 15. Next, using the Highlight Range tool, select several good oscillations. . 97

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16. Using the Apply Selected Curve tool, apply a ”sine” fit to highlighted data and record the angular frequency, ! , in Data Table 1. 17. Add additional mass and repeat for five additional times, recording the period and angular frequency of oscillation each time. Make sure to keep the displacement o the mass, x , the same each time. 18. NOTE: Be sure to use the ”Delete Last Run” button on the bottom of the screen next to the frequency adjustment, or the sine fit from your previous run will still show. 19. NOTE: Make sure to NOT over-displace the spring so that it bounces the masses o ↵ the hanger or add too much mass as to over-stretch the spring. 20. Repeat these steps for a total of three di ↵ erent springs. 98

Data Table 1 Color of spring: Mass of spring: Displacement of spring: Angular Frequency Trial m total (kg) 1/ m total (1/kg) Period, T (s) Measured ( ! m ) Calculated ( ! c ) ( ! m ) 2 1 2 3 4 5 Color of spring: Mass of spring: Displacement of spring: Angular Frequency Trial m total (kg) 1/ m total (1/kg) Period, T (s) Measured ( ! m ) Calculated ( ! c ) ( ! m ) 2 1 2 3 4 5 99

Color of spring: Mass of spring: Displacement of spring: Angular Frequency Trial m total (kg) 1/ m total (1/kg) Period, T (s) Measured ( ! m ) Calculated ( ! c ) ( ! m ) 2 1 2 3 4 5 11.3 Data Analysis 1. Using the Period, T , and Equation 33, calculate the angular frequency, ! c , and record in each of the data tables in the Data Collection section. 2. How does the calculated ! c compare the to the measured ! m ? 3. Calculate and record ( ! m ) 2 in each of the data tables in the Data Collection section. 4. Calculate 1 /m total and record in each of the Data Tables. 5. Using Excel, plot your dependent and independent variables. 6. Using the value of the slope, from the best-fit line for each graph, determine the value of the springs constants for each spring and record in the Data Analysis Table . 7. Show your Lab Instructor your graph(s) for each spring and have your instructor sign here: . 8. Compare your experimental value with the given values obtained from your Lab In- structor. Data Analysis Table Spring Color Spring Constant, k ( N m ) Given k % Di ↵ erence 100

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9. Choose two consecutive data points from one of your springs. 10. Record the required information. Color of spring: Trial m total (kg) Period, T (s) 1 2 11. Using your data, show that the angular frequency, ! is dependent on the square root of the spring constant over the total mass (Equation 34). Show your work. Lab Instructor Signature Score / 10 points 101

11.4 Lab 11 - Lab Summary Question This question is an individual assignment and not a group discussion. Please write your full name, ID number, and lab section as indicated and use the space provided to write a complete answer to the question. Name: ID Number: Lab Section: Question: For the following two scenarios agree or disagree and give evidence as to why. 1. A worker, installing a screen door with a spring used to shut the door after it is opened, choose a large spring constant so the door would close slowly. 2. A car company, using spring shocks, chose to use springs with a small spring constant to help cushion the card’s ride when going over bumps on the road. Answer: 102

Lab 11 Mass-Spring System CPI F23 (1)

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