Gravitational & Rotational Energy

For a system involving only conservative forces, the total mechanical energy of the system will be conserved. Mathematically, we can say:

The purpose of this experiment is to confirm the conservation of mechanical energy for a conservative system involving both translational and rotational motion. Figure. 1: Experimental Setup

The equipment for this lab includes a rotational motion apparatus, a vertical pulley, an electric balance, a photogate and Smart Timer, a stopwatch, and a 500 g mass. Our experimental setup is pictured in Figure 1. The rotational dynamic apparatus is comprised of a primary rotating disc with a mounted step pulley. A string is wound around the top level of the step pulley, draped over the vertical pulley, and connected to a 500 g mass. The hanging mass generates a torque on the disc, causing it to rotate.

In this lab, we will start with the 500 g mass just below the vertical pulley and allow it to fall to the ground. We will show that the total change in kinetic energy of the system is equal to the total change in the potential energy of the. The change in potential energy of the system is given by:

The change in kinetic energy is given by:

where ωf is final rotational velocity of the disc and vf is the final translational velocity of the falling mass. Therefore, conservation of mechanical energy finds:

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Name:

Lab Partners:

Procedure: 1. Measure the mass of the disc:

md = g

2. Measure the diameter of the step pulley with the digital caliper (make sure to measure its diameter on the inside track) and divide it by 2 to get its radius:

rp = cm

3. Measure the outer diameter of the primary disc of the rotational motion apparatus and divide by two to gets its radius. Then, flip it over and measure the diameter of the hole in its center, divide by two, and record its radius:

rout = cm

rin = cm

4. Place the disc back on the apparatus. Give it a turn to ensure it rotates freely. Run the thread, with the weight hanger attached, over the vertical pulley. Attach 500 g of mass to the string. Ensure that the string is of sufficient length to allow the mass to hit the ground. If this is not the case, notify your instructor. Now, rotate the disc until the mass hangs just below the vertical pulley. Make sure to secure the disc so that it does not begin to rotate. Record the height of the bottom of the 500 g mass using the meter stick

hi = cm

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5. Record Time. Turn on the Smart Timer. The photogate should be connected to input channel 1. Set the timer to “Time/Fence” mode. Press “Start/Stop” before the first trial.

When the disc is unsecured, the thread will generate a torque on the disc causing it to rotate. In “Time/Fence” mode, the Smart Timer will start the clock when the card first intercepts the beam of the photogate. It then records the elapsed time since this first trigger for every subsequent interception of the beam.

You will now allow the disc to rotate from rest. You are to allow the mass to travel all the way to the ground. Using your stopwatch, one member of your group should measure the time of flight, ∆ t, for the 500 g mass. At the same time, another member of the group should press the “Start/Stop” button on the Smart Timer the moment the mass hits the floor. At this point, pressing the “Select Measurement” button will allow you to see the times measured by the Smart Timer. Record these data below in order of increasing time, along with the time of flight, in the tables below.

Smart Timer ti (s)

t2 Stop Watch - t3 ∆ t (s) t4 t5

t 6 t7 t8

Analysis

1. Angular Velocity. The average angular velocity, ωi, during each of these intervals is given by the angular distance traveled by the disc, 2 π radians, divided by the elapsed time for the full rotation. Calculate each ωi and record the data in the table below.

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ωi (rad/s)

ω1

ω2

ω3

ω4

ω5

ω6

ω7

ω8

2. Scale Time to Start at Zero. Offset the measured time to start at zero by subtracting your initial time from each time measurement.

t s,i (s)

t s,1 0

t s,2

t s,3

t s,4

t s,5

t s,6

t s,7

3. Assign time to each angular velocity. The

Smart Timer takes readings each time the disc t s,1 = 0 rotates—thus at equal distances, but changing time intervals. In order to determine angular t s,2 – t s,1 acceleration, we need to assign a time to each angular velocity. Assume that the average t angular velocity between any two time readings s,2 occurred at the halfway point between them. t s,3 – t s,2 This “average time” is: t s,3

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tavg (s)

t1 avg

t2 avg

t3 avg

t4 avg

t5 avg

t6 avg

t7 avg

4. Angular Acceleration. Using MS Excel create a plot of the angular velocity vs. tavg for each trial. The mathematical relation ship between angular velocity and time is given by the following equation.

A plot of ω vs. t avg should yield a line with a slope equal to the angular acceleration, α. Once your plot is complete, have Excel determine the best fit slope ( α) through the method of least squares. Record your α below.

α = rad/s 2

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5. Calculate the final angular velocity of the disc:

ωf = rad/s

6. Calculate the expected moment of inertia of the disc using the formula

2 Iexp = g cm

7. Calculate the | ∆ U |:

| ∆ U | = erg

8. The final translational velocity of the falling mass is related to the final angular velocity of the disc in the following manner:

Therefore, the change in kinetic energy can be written:

Use this equation to find the | ∆ K|

| ∆ K | = erg

9. Compute the percent difference between the change in potential energy and the change in kinetic energy.

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