EXPERIMENT

Rotation

Objectives

By the end of the experiment, the students should be able to:

ˆ verify the conservation of energy and angular momentum for a system of disks
rotating about the central axis.
ˆ calculate the net torque, kinetic energy, angular momentum, and the work done by
friction for a disk that is rotating about its central axis.

Introduction

Rotations play an important part in understanding the behaviour of a rotating wheel, a

spinning top, and the motion of the planets around the solar system. The concepts of
conservation of energy and angular momentum simplify the analysis of problems where
rotations are involved. In this experiment, the students will verify the conservation of
energy and angular momentum for a disk that is rotating about its central axis. Along
the way, the students will learn how to calculate the net torque, kinetic energy, and the
work done by friction. Also, conservation of angular momentum will be tested for the
perfectly inelastic collision of a system of disks.

Single disk rotating about its central axis

Consider a uniform disk that is rotating about its central axis as shown in Figure 1. The
friction between the disk and the shaft is causing heating, thus, slowing down the rotation
of the disk until it stops.

Torque
The torque ⃗τ is the tendency of a force to rotate an object about some axis. By definition,

⃗τ = ⃗r × F⃗ (1)

where F⃗ is the applied force and ⃗r is the displacement vector from the axis (pivot) to the
point where the force is applied. The net torque that is experienced by an object causes
an angular acceleration α⃗ that is defined through Newton’s second law for rotations:
X
⃗τ = I⃗
α. (2)

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1st Semester, A.Y. 2021–2022 Physics 71.1

Figure 1: A disk that is rotating with an angular speed ω about its central axis

In equation 2, I is known as the moment of inertia about the axis of rotation.

For a uniform annular disk with mass M , inner radius R1 , and outer radius R2 that
is rotating about its central axis, the moment of inertia is given by

1
Idisk = M R12 + R22 .


(3)
2
The net torque on the rotating disk is therefore the product of equation 3 and the angular
acceleration.

Conservation of energy
The total energy of the system is the sum of the mechanical energy and the internal
energy. The mechanical energy is defined as the sum of the kinetic and the potential
energy. For a disk that is rotating at a constant height, the change in mechanical energy
is due in part only to the change in the kinetic energy of the system:

1
∆Emech = ∆KE = Idisk (ωf2 − ωi2 ). (4)
2
In equation 4, ωi and ωf are the initial and final angular velocities, respectively.
The work done by friction on the disk increases the total internal energy of the system.
Using equation 1, the torque applied by friction which is tangent to the inner radius R1
is given by τ = f R1 where f is the magnitude of friction. Under the constant friction
assumption, the work done by friction can be written as

Wfriction = −f R1 |∆θ| = −τ |∆θ| (5)

where ∆θ is the angular displacement of the disk.

The change in total energy is the sum of the changes in the mechanical energy and
the internal energy. If the there are no external forces acting on the system, then the
total energy must be conserved:

∆E = ∆Emech − Wfriction = 0. (6)

Perfectly inelastic collision of disks

Consider a first disk that is rotating about its central axis. A second disk is released from
rest on top of the first disk (Figure 2) and collides with the first disk, thus, making a
composite two-disk system that is rotating with some angular velocity.

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Physics 71.1 1st Semester, A.Y. 2021–2022

Figure 2: Another disk is dropped from rest onto a disk that is rotating about the central
axis

⃗ for a system of rotating disks is defined as

The total angular momentum L
X
⃗ =
L Ii ω
⃗i (7)
i

where ω⃗ i is the angular velocity for each disk. When a net torque is applied for a time
interval ∆t, the rotational impulse J⃗ experienced by the system is given by
X
J⃗ = ⃗τi ∆t. (8)
i

When the net torque is zero, then the rotational impulse-angular momentum theorem

J⃗ = ∆L
⃗ (9)

⃗ = 0).
suggests that the angular momentum is conserved (∆L

Materials

The experiment requires the use of the fol-

lowing equipment:

ˆ GoDirect rotary motion sensor

ˆ Disks

ˆ Ruler

ˆ Digital weighing scale Figure 3: Disks

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1st Semester, A.Y. 2021–2022 Physics 71.1

Procedure (Set A)

Calculation of the moment of inertia of the disks

1. Measure the inner radius R1 and outer radius R2 of each of the disks using a ruler
and record in Table W1. (See videos Disk Intro.mp4 and Ruler Intro.mp4)

2. Measure the mass of each of the disks and record in Table W1.

3. Calculate the moment of inertia for each disk using equation 3 and complete Table
W1. Don’t forget to convert units.

Note: The dimensions and masses of the disks are already provided in the Data Sheet.
Morover Disk 2 will not be used in this version of the manual.

Verification of the conservation of energy

The supplemental material Rotation Spinning Disk.mp4 shows how this section of
the experiment is performed.

1. Connect the rotary motion sensor to the Graphical Analysis 4 app.

2. Attach disk 1 to the rotary motion sensor and tighten the screws as shown in Figure
1.

3. Start the data collection by pressing the Collect button on the app and give the
disk a spin. The provided clip Spinning Disk Graph.mp4 shows how the θ − t
and ω − t plots are generated.

Figure 4: Giving disk 1 a spin. From video Rotation Spinning Disk.mp4

.

4. Select a part of the graph where the angular velocity vs time is linear. Take note
of the times ti and tf , the angular positions θi and θf , and the angular speeds ωi
and ωf and complete Table W2.

5. Complete Table W3 using the data from Table W2.

6. Calculate the change in mechanical energy, the work due to friction, and the percent
error |(∆Emech − Wfriction )/∆Emech | × 100%. Complete Table W4.

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Physics 71.1 1st Semester, A.Y. 2021–2022

Verification of the conservation of angular momentum

The supplemental material Rotation Drop Disk.mp4 shows how this section of the
experiment is performed.

1. See Figure 2 for a visual representation of this section of the experiment.

2. Record the moment of inertia of disk 1 as Ia and the moment of inertia of disk 3 as
Ib in Table W5.

3. Attach disk 1 to the rotary motion sensor.

4. Start data collection and give disk 1 a spin.

5. Drop disk 3 from an arbitrary height onto disk 1 and observe the sharp change in
the θ − t and ω − t graphs. The clip Drop Disk Graph.mp4 shows how these
plots are generated.

Figure 5: Dropping a stationary second disk on top of an initially rotating disk. From
video Rotation Drop Disk.mp4
.

6. Take note of the angular velocity just before the collision and record this as ωa,i in
Table W5.

7. Take note of the angular velocity right after the collision and record this as ωb,i in
Table W5.

8. Calculate the initial angular momentum Li , the final angular momentum Lf , and
the percent error |(Li − Lf )/Li | × 100% and complete Table W5.

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1st Semester, A.Y. 2021–2022 Physics 71.1

Addendum

Verification of the conservation of energy

Figure 6: Sample ω vs time, and θ vs time plots.

Shown in Figure 6 is a plot of the expected angular speed versus time and its corre-
sponding angular displacement versus time curve for the disk. Once the disk spins freely,
the speed of the disk decreases linearly in time due to friction. A sample data acquisition
performed using the Graphical Analysis 4 app is shown in the Spinning Disk Graph.mp4
video.

Verification of the conservation of angular momentum

Figure 7: Sample ω vs time plot for two disks that undergo a perfectly inelastic collision.

Shown in Figure 7 is a plot of the expected angular speed versus time plot for two
disks that undergo a perfecctly inelastic collision. The first disk initially rotates at a
specified angular speed. The second disk is then dropped on top of the first disk. After
the collision, the two disks are now in contact and rotate with the same final angular
speed. A sample data acquisition performed using the Graphical Analysis 4 app is shown
in the Drop Disk Graph.mp4 video.

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Physics 71.1 1st Semester, A.Y. 2021–2022

Name: Date:
Instructor:

Data Sheet (Set A)

In the data sheet, the students are expected to:

ˆ Fill Tables W1 to W5.

ˆ Calculate uncertainties only for Icm . Uncertainties are not required for other quan-
tities.

ˆ Attach sample calculation.

Moment of inertia of the disks

Table W1: Calculation of the moment of inertia of the disks

disk inner radius outer radius mass Icm (kg·m2 )

1 0.55 ± 0.02 [cm] 8.80 ± 0.02 [cm] 102.8 ± 0.1 [g]
2 — — — —
3 2.70 ± 0.02 [cm] 8.70 ± 0.02 [cm] 256.4 ± 0.1 [g]

Calculation of the net torque

Table W2: Data for calculating the net torque on disk 1

Trial ti (s) θi (rad) ωi (rad/s) tf (s) θf (rad) ωf (rad/s)

1 4.00 [s] -181.78 [rad] -26.36 [rad/s] 9.35 [s] -309.08 [rad] -21.67 [rad/s]
2 0.55 [s] -85.71 [rad] -29.57 [rad/s] 4.30 [s] -189.65 [rad] -25.91 [rad/s]
3 1.20 [s] 90.58 [rad] 24.97 [rad/s] 5.40 [s] 188.13 [rad] 21.74 [rad/s]

Table W3: Calculation of the net torque on disk 1

Trial ∆ω (rad/s) ∆t (s) |α| (rad/s2 ) |τ | (N·m)

1
2
3

Verification of conservation of energy

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1st Semester, A.Y. 2021–2022 Physics 71.1

Table W4: Data for verification of conservation of energy

Trial Ki (J) Kf (J) ∆Emech (J) Wfriction (J) % Error

1
2
3
Average percent error

Verification of conservation of angular momentum

Table W5: Data for verification of conservation of angular momentum

Trial Ia (kg·m2 ) Ib (kg·m2 ) ωa,i (rad/s) ωb,f (rad/s) Li (kg·rad·m2 /s) Lf (kg·rad·m2 /s) % Error
1 -25.88 [rad/s] -6.31 [rad/s]
2 -30.52 [rad/s] -8.60 [rad/s]
3 26.35 [rad/s] 8.99 [rad/s]

Questions

Answer the following questions in no more than two sentences.

1. A uniform disk with moment of inertia at the center of mass, I1 , is rotating with an
initial angular speed ω1 about a frictionless shaft through its center. Another disk
of moment of inertia at the center of mass, I2 , is dropped from rest on top of the
first disk through the same shaft. Give the angular speed of the composite two-disk
system assuming that the collision is perfectly inelastic.

2. Is the total energy of the rotating disk conserved in the experiment? Justify your
answer using the data in Table W4.

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Physics 71.1 1st Semester, A.Y. 2021–2022

3. In the collision of disks, the system is not isolated because of existence of friction.
How is it possible that conservation of angular momentum is still valid to a relatively
good accuracy?

4. In most cases, Disk 3 will not be concentric with the other disk after it is being
dropped. Will the final speed of this configuration be greater than or less than the
speed in theoretical scenario (all 3 are concentric)? Explain.
Hint: Parallel axis theorem is needed to solve the moment of inertia of Disk 3 about
the axis of rotation.

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1st Semester, A.Y. 2021–2022 Physics 71.1

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