lab Rotational Motion

Lab Instructions V0.0 June 20, 2018

Rotational Motion

.

1 Objective To experimentally study rotational motion of a rigid object and to determine its moment of inertia.

2 Overview Figure 1 gives an overview of the experiment. The rigid object under study is a turntable and axle system (Pulley platter) with moment of inertia I0, on which other rigid objects, like a solid disk, thick ring, rectangular bar, with moment of inertia Iadd, can be placed to create a composite object with a moment of inertia, I = I0 + Iadd. A thin string is wrapped around the middle of platter and passed over a pulley. The other end of the string hangs vertically with a small mass, m, tied to it. When the mass is released, it accelerates vertically down and the axle-turntable system rotates with an angular acceleration. Let a and α be the magnitudes of acceleration of the mass and that of the angular acceleration of the axle-turntable system, respectively. Recall, from lecture that the magnitude of the tangential acceleration, aT , of a point on the circumference of the axle is related to α by, aT = αr0, where r0 is the radius of the axle. Since the string unwinds without slipping, so the magnitude of its acceleration, and hence, that of the mass attached to it, satisfies a = aT . So,

a = αr0 (1)

Applying Newton’s second law to m (all forces on m are vertical),

mg − T = ma, (2)

where T and g are the magnitude of the tension in the string (assumed massless) and the magnitude of the acceleration due to gravity, respectively. The turntable-axle system then experiences a torque of magnitude,

τ = r0T, (3)

which in turn, gives it an angular acceleration. The rotational analog of Newton’s second law states,

For use by the physics department, Ohlone College, Fremont, CA.

Lab Instructions V0.0 June 20, 2018

τ = Iα, (4)

where,

(5) I is Inertia

From Equations 1 through 4, we can solve for a to obtain,

a = g

1 + I mr20

(6)

In your lab report, make sure to derive Equations 2, 3 and 6.

In this experiment, m is chosen to be small, so that a, in turn, is small enough that it can be easily determined by utilizing a stopwatch to measure the time, ∆t, it takes m, starting from rest, to the descend through a measure distance, ∆y. Since, the motion of m is slow, ∆t can simply be measured by a stopwatch. From kinematics of 1D motion with uniform acceleration, starting from rest, we get

∆y = 1

2 a∆t2 (7)

With a determined from Equation 7, I can be readily obtained by employing Equation 6.

Figure 1: Overview

3 Apparatus

1. Rotational Motion turntable (Pulley platter).

2. A solid disk, thick ring, rectangular bar that can be place on the turntable.

3. Two clamps.

4. One metal rod.

For use by the physics department, Ohlone College, Fremont, CA.

Lab Instructions V0.0 June 20, 2018

5. One pulley.

6. A book or some other object to raise the height of the turntable.

7. A thin thread or string.

8. A small mass of few grams with an eye or hook to tie to a string.

9. A mass hanger and slotted masses 10 − 50 g.

10. A meter stick and a stop watch.

Figure 2: Apparatus

Figure 3: Experimental Setup

4 Precaution

• BE CAREFUL TO NOT DROP THE TURNTABLE, HOOP OR DISK, AS THEY ARE HEAVY AND CAN CAUSE INJURY/DAMAGE.

• MAKE SURE THAT THE STRING IS HORIZONTAL AND REMAINS TANGEN- TIAL TO THE AXLE’S SURFACE AS IT UNWINDS.

For use by the physics department, Ohlone College, Fremont, CA.

Lab Instructions V0.0 June 20, 2018

Figure

5 Part 1: Determine I0

1. For this part of the experiment, utilize the turntable-axle as-is; don’t place any other object, hoop, disk, etc., on the turntable.

2. Follow Figure to set up the experiment.

3. There is a small pin on the axle. Securely tie one end of the string to it before winding it on the axle.

4. Make sure that the string remains horizontal as it passes over the pulley, and it remains perpendicular to the radius of the axle. Why? You might have to adjust the position and height of the turntable, by placing a book or box under it, etc., to accomplish this.

5. Make a small loop at the other end of the string and then hook the hanging mass m on to it. Experiment to find a suitable value of m (∼ few grams) that will descend slowly as you release it, taking about 10 to 15 seconds to fall to the floor. Record m .

For use by the physics department, Ohlone College, Fremont, CA.

Lab Instructions V0.0 June 20, 2018

Using a software we will get velocity and time for each object (check the excel sheet)

5.1 Calculations

1. Calculate the acceleration from velocity vs time graph for Pulley platter + objects. Then, utilize Equation 6 to calculate the experimental value of I0. Record it in Table below:

For use by the physics department, Ohlone College, Fremont, CA.

acceleration [m/s 2 ] Rotational inertia (I0)[kg·m

2 ]

Pulley platter

acceleration [m/s 2 ]

Rotational inertia (I= I0+ Iadd)[kg·m 2 ]

Pulley platter

+ solid disk

Pulley platter

+ thick ring

Pulley platter

+ rect bar

Solid disk 2

2

1 I  M disk R

Thick ring I  1

2 M ring a2  b2

Rectangular bar I  1

12 M bar l2  w2

Theoretical rotational inertia values (Iref)(various shapes):

Lab Instructions V0.0 June 20, 2018

6 Part 2: Determine Iadd

1. Repeat the experiment, but this time, add another object, disk or hoop to the turntable.

2. Now, the hanging mass will need to be increased. Again, you need to experiment to find a suitable value that will descend down slowly. It will likely be in the 50 g to 100 g range. You can use the mass hanger and slotted masses. Note: The mass will be the sum of the mass of the hanger and slotted masses. Record m in the appropriate place in Table 1.

3. Either measure, or read-off from the label, the mass, Madd, and radius, Radd, of the add- on object. Record it in Table 3.

6.1 Calculations

1. Compute the moment of inertia of just the add-on object, Iadd, from I = I0 + Iadd.

2. Utilize Madd and Radd, to compute the theoretical moment of inertia, Iref of the add-on object. Look up the expression or derive it!

3. Record all results in Table

7 Error analysis and discussion

1. Discuss possible sources of error to the experimental value of I0 and Iadd?

2. How would you measure the torque due to friction in the axle?

3. Why is it important for the string to be horizontal?

8 Conclusion

Briefly summarize your findings.

For use by the physics department, Ohlone College, Fremont, CA.

Description Experimental Rot

Inertia [kg·m2] Theoretical Rot

Inertia [kg·m2] % error

Disk alone

Ring alone

Bar alone