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Rotational Inertia of a Point Mass

Saddleback College Physics Department, adapted from PASCO Scientific PURPOSE The purpose of this experiment is to find the rotational inertia of a point mass experimentally and to verify that this value corresponds to the calculated theoretical value. EQUIPMENT

- pulley - paper clips (for masses < 1 gram) - calipers - masses and weight hanger set - 2 photogate timers with 3 photogate heads - triple beam balance - meterstick - 300 g (black, square) mass - rotating platform - “A” base - thread/string

SKETCH

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Figure 2.3 The set-up shown above, with the photogate in Pendulum mode, enables one to determine whether the hanging mass is falling at an approximately constant rate.

Figure 2.4 When arranging the two photogates as shown to the left, be sure the string and hook (at the top of the weight hanger) do not break the photogate beam.

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THEORY Theoretically, the rotational inertia, I, of a point mass is given by 2

MRI = , where M is the mass and R is the distance the mass is from the axis of rotation. To find the rotational inertia of a point mass experimentally, a known torque is applied to the object and the resulting angular acceleration is measured. Since !" I= , rearranges to

!

"=I (1) where

ra=! (2)

and rT=! (3) (where r = radius of cylinder about which the thread is wound and T is the tension in the thread when the apparatus is rotating.)

Substitute equations (2) and (3) into equation (1) above and write the result in the box below. (4) Apply Newton’s Second Law to the hanging mass, call it m for now, and then solve the resulting equation for the tension in the thread, T. Substitute this value for T into equation (4) above and you have a formula yielding the experimental rotational inertia in terms of the linear acceleration of the mass, m. Call this new equation (5) and write it below.

(5) The linear acceleration, a, of the mass, m, can be found with the kinematic equation below:

2

2

1attvy +=!

o (6) where

ov is the initial velocity of the hanging mass (zero),

and t is the (average) time for the hanging mass to fall from rest through a displacement y! .

NOTICE: You will be solving the above equations for two different cases to find the

experimental rotational inertia of the point mass. To find masspo

Iint

you must find

apparatusmasspoI

&int then subtract out the rotational inertia of the apparatus alone (i.e.

with no point mass). The mathematical relationship is seen below:

masspoI

int=

apparatusmasspoI

&int -

aloneapparatusI (7)

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PROCEDURE

Equipment Setup 1) Leveling the Rotating Platform:

This may take 30 minutes to do correctly, but it is time well spent! Before leveling the apparatus you MUST set entire apparatus in the position shown in Figure 2.4. Notice that the mass hanging off the pulley cannot hit the table and the string wound around the cylinder must pass straight over the pulley.

ONCE APPARATUS IS LEVEL, YOU CANNOT MOVE IT!

2) Set up your apparatus as shown in Figures 2.1 and 2.2 (above), making sure the

hanging mass almost reaches the floor when the string is completely unwound.

Part I: Measurements for the Theoretical Rotational Inertia

1) Determine the Mass of the square black mass, M (the point mass) and record it in table 2.1.

2) Attach the square black mass to the track on the rotating platform at a large radius.

3) Measure the distance from the axis of rotation to the center of the square mass and record this radius, R, in Table 2.1

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Theoretical Rotational Inertia Mass, M

Radius, R

Table 2.1 Part II: Measurements for the Experimental Rotational Inertia ACOUNTING FOR FRICTION Because the theory used to find the rotational inertia experimentally does not

include friction, it will be compensated for in this experiment by finding out how much mass over the pulley it takes to overcome kinetic friction and allow the mass to drop at a constant speed. Then this “friction mass” will be subtracted from the hanging mass that was used to accelerate M and the resulting mass will be m, see calculations.

1) To find the mass required to overcome kinetic friction, place a photogate (set on “PENDULUM”) so that the rotating platform will break the beam and time one complete cycle of the rotating platform.

2) Tie the string through the hole in the middle cylinder and wind it evenly about that cylinder.

3) Place just enough mass over the pulley (using paperclips) so that the rotating platform moves at a constant angular speed. You might need to lightly nudge the hanging mass or pulley to help the system overcome static friction. Just eyeball the speed at first then when you think the speed is nearly constant, use the photogate (set on Pendulum mode) to time a revolution early on and to time a revolution toward the end of the motion (see Figure 2.3 for set-up). Clearly these two periods should be the same if a constant angular speed is attained. Try to get the periods to match to three significant figures. Record the friction mass in table 2.2, under the appropriate column.

Finding the Acceleration of the Point Mass and Apparatus

1) You will not use the above photogate until you repeat the procedure with the “Apparatus Alone.” Carefully clamp/tape the other two photogate heads onto a vertical stand about 50+ cm apart and connect them to a photogate timer set on “PULSE” and “ ms1 ” (which means the precision of the photogate timer is to the nearest millisecond, but the units of the reading are still seconds). See Figure 2.4 for set-up. The top photogate beam must be broken by the hanging mass immediately when it begins to fall, since the initial velocity of the hanging mass is assumed to be ZERO in equation (6) of the theory. Measure the separation between the photogate heads, y! , and record it in table 2.2. Make sure the string suspending the hanging mass does not pass through the beams of the photogate heads, as this can break the beam prematurely! When the photogate timers reach 20 seconds they reset to 10 seconds and then continue to count every 10 seconds thereafter. Unfortunately they are intended to count shorter intervals. Do a practice run and just watch the photogate timer to learn how to correctly read and record the time of fall.

2) To find the acceleration, place about 50 g (record the hanging mass in table 2.2 under the appropriate column) over the pulley and wind the thread around the

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platform. Using the vertical photogates, time the hanging mass as it falls through the photogates and record the time in the appropriate column in table 2.2. Do this for a total of 15 trials.

Measuring the Radius, r Using calipers, measure the diameter of the cylinder about which the thread is wrapped and calculate the radius. Record this in Table 2.2. Finding the Acceleration of the Apparatus Alone Since in the Part II procedure “Finding the Acceleration of the Point Mass and

Apparatus,” the apparatus is rotating as well as the point mass, it is necessary to determine the acceleration and rotational inertia of the apparatus by itself so that this rotational inertia can be subtracted from the total, leaving only the rotational inertia of the point mass.

1) Take the point mass off the rotational apparatus and repeat the Part II procedure

“Finding the Acceleration of the Point Mass and Apparatus” for the apparatus ALONE!

2) Record the data in Table 2.2 under “Apparatus Alone.”

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Table 2.2

Point Mass & Apparatus Apparatus Alone

Friction Mass

Hanging Mass

Mass, m (for use in equations)

Radius, r (of cylinder)

Acceleration, a (of hanging mass)

Distance between Photogates, y!

Time, t (to fall through photogates) Point Mass & Apparatus Apparatus Alone

1t

2t

3t

4t

5t

6t

7t

8t

9t

10t

11t

12t

13t

14t

15t

Average time, t

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Calculations 1) Subtract the “friction mass” from the hanging mass used to accelerate the

apparatus to determine the mass, m, to be used in the equations and record in Table 2.2.

2) Use the average time of fall to determine the acceleration of the system from equation (6). Record the acceleration in the appropriate column of Table 2.2.

3) Calculate the experimental value of the rotational inertia of the point mass and apparatus together and record in Table 2.3.

4) Calculate the experimental value of the rotational inertia of the apparatus alone. Record in Table 2.3.

5) Apply equation (7) in the theory to get the rotational inertia of the point mass alone. Record in Table 2.3.

6) Calculate the theoretical value of the rotational inertia of the point mass. Record in Table 2.3.

7) Compare the experimental value to the theoretical value. Record in Table 2.3.

Results

apparatusmasspoI

&int

aloneapparatusI

masspoI

int (Experimental Value)

masspoI

int (Theoretical Value)

% Difference in masspoI

int

Table 2.3