Exp 3 - Rolling Disc and Axle

8/9/2019 Exp 3 - Rolling Disc and Axle

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In this experiment you will measure I for a disk mounted on an axle. The axle can be thought

of as a very thick disk and you can use the same expression to compute Idisk and Iaxle. The

total I of the disk + axle is the sum of these two.

Diagram 1

(2)

In this experiment, you will determine I in two ways:

1) Measure the masses and radii of disk and axle and then compute I from the formula

above.

2) Compute I by timing the wheel as it rolls down inclined rails and using the principle

of conservation of energy.

Consider the wheel, consisting of disk and axle, rolling down an inclined set of rails after

starting from rest at the top, like so:

Diagram 2


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The total energy at any time is the sum of the translational kinetic energy, the rotational

kinetic energy, and the gravitational potential energy.

. (3)

where: M is the total mass of disk + axle,

v is its translational speed ,

ω is its angular velocity, and

h is the height of the centre of mass.

Initially, the wheel is at rest at height ho, so its initial kinetic energy (both translational and

rotational) is zero and its total energy is all potential energy.

(4)

When the wheel reaches the bottom of the rails, h=0, and the energy is all kinetic energy:

(5)

where: vf is the final translational speed and ωf is the final angular speed.

Because the rolling friction is very small, we can assume that the total energy is constant asthe disk rolls down the rails, and so the initial energy is equal to the final energy.

. (6)

For an axle or wheel that rolls without slipping, the angular velocity ω and the translational

speed v are related by:

. (7)

Note that here and equation (8) below, r is the radius of the axle, NOT the radius of the big

disk!

Using equations (6) and (7), one can find I in terms of M, r, g, vf , and ho.

(8)

You are asked to derive equation (8) in the questions. You will use this expression to

determine I using two different choices of the initial height ho, thus yielding two new valuesof I that you can compare with I calculated from equation (2).


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Methodology

Part 1: Measurement of I from dimensions and masses of disk and axle

In this experiment, it is a good idea to use centimeters and grams, rather than meters and

kilograms for all your measurements. This is because the moment of inertia of our disks turnsout to be a very small number (roughly 10-3 kg× m2) and it is a little awkward to work with

small numbers. If you use cgs (centimeter-gram-second) units, you must be consistent and

always use cgs units, so use gravity g = 979.6 cm/s2 (not 9.796m/s2).

Gently slide the axle out of the disk and weigh both separately to find their masses. Measure

their diameters to find their radii, r for the axle and R for the disk. At this stage you do not

need to know r very precisely, but you will in part 2, so measure the diameter of the axle very

carefully three or four times with the calipers. Use the average of your measurements and

estimate the uncertainty in r. (If you don't know how to use the calipers, ask your instructor.)

Using equation (1), find Idisk and Iaxle separately and then compute . (Is Iaxle

significant, compared to Idisk , or can it be ignored?) In using equation (1) to compute Idisk , we

are making a small error by ignoring the hole in the center of the disk. Compare the "missing

mass" of the hole to the mass of the axle to determine whether this omission is significant.

Result for part 1 consists of 3 sets of measurements of d disc, daxle, mdisc, maxle, R disc, r axle, and

the average values. The calculation involves calculation for I using equation (2).

Part 2: Measurement of I using energy conservation.

1. Place the inclined plane apparatus on a level surface and ensure that the top surfaces of

the two rails are levelled. Wipe off any grease and dirt, which may be on the top of the

rails.

2. Measure from the table top to the underside of the front rail at each end (a, b) and

record the height in table. Measure the overall length of the rail (Lo) and note the value.

Refer Diagram 3.

3. Raise the movable end of the rails to the lowest level so that the rails are almost

levelled. Place the large disc on the rails at the support end and mark its position.

* Hold on the disc near the start block so that once it’s released it should roll freely

along the rail

4. Using the measuring tape, measure about 0.7m along the rail from the centre pivot of

the disc axle and fix the stopper block so that the linear distance travelled by the disc is

0.7m along the rail. Refer Diagram 4.

5.

Remove the disc and raise end of the rail for about 25mm. Measure the heights a and b from the table top.


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6. Place the large disc on the high end of the rails and line up the axle perpendicular to the

rails. Hold it while preparing to use the stopwatch.

7. Measure the time taken for the released disc to roll itself 0.7m down the inclined rails.

Repeat this procedure 3 times and record the average time taken into the table.

8.

Raise the support by another 25mm height and repeat the step 5 and 6. And repeat step

7 twice.

9. Repeat the whole experiment for the small disc.

10. Plot the graph of log t against log H for both of the disc on graph paper.

11. Refer to Appendix 1 for the detail calculation.

Diagram 3

Diagram 4

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Questions

1. Starting with equations (6) and (7), derive equation (8). Be sure to show all your steps.

2.

What are the units of moment of inertia I? [Give the answer both in cgs (centimeter-gram-second) units and in MKS (meter-kilogram-second) units.]

3.

For a wheel that rolls without slipping, if you are given its angular speed ω, and its

radius r, what is its translational speed v?

4. In part 2 of this lab (equation 8) you will need to know vf , the final speed of the wheel.

Why is it incorrect to find vf by simply computing d/t? What is the correct expression

to find vf ?

5.

In part 2 of this lab (Eq'n 8 again) you will need to know r. Draw a quick sketch of the

wheel, and clearly indicate what "r" is.

6. Equation 1.8 in the appendix involves g, the acceleration of gravity. This seems tosuggest that you would get a different value for I if you conducted the same experimenton the moon, where g is different. But the definition of I does not depend on location.Like the mass m, the moment I of an object is the same on the moon as on the Earth oranywhere else. So how do you explain the presence of g in Equation 1.8?


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Exp 3 - Rolling Disc and Axle

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