Penn State University
Physics 211R: Lab
–
Torque and Angular Momentum
1
Physics 211R: Lab Report Template
Torque and Angular Momentum
(Type in this document and print
these pages
at the end of the laboratory)
NOTE: Remember to SAVE this template as a Word
document before starting the lab. Always
remember to save your template periodically throughout the lab.
A maximum of three stu
dents will be allowed per group without prior instructor
permission.
All the members of the group must participate in the activ
ity. If a student is not
participating (even when present) s/he may receive a score of zero in the activity.
Students arriving 10 minutes or more past start will not be admitted.
This activity must be returned at the end of the lab period. All the students
completing
the activity must be present when handing this to the laboratory instructor; a student
not present at this time may not get credit for the activity.
Writing the name of a
person not present is not permissible and may result in a potential acade
mic integrity
violation being processed.
After you receive the graded report back, you should make a copy of the front page
(this page) and keep it for you records. This will serve as evidence of your grade for
this activity.
You are responsible for checki
ng your grade an
d report any mistakes to your
laboratory instructor
within two weeks after the activity.
Date: ________________
Enter your name as it appears in your PSU registration, no nicknames please.
Name:
Section #
Name:
Section #
Name:
Section #
Clean Up Check:
After you finish working and completing the lab report, you need to clean and organize
your working area. Then call one of your laboratory instructors who will check your area,
initialize below and take the lab report. All the members of the group must be
present at
that time. If you leave the lab before your laboratory instructor performs the check up,
you will be deducted 5 points
from your score for this lab report.
Laboratory Instructor Initials:
_______
Score:
_______
Penn State University
Physics 211R: Lab
–
Torque and Angular Momentum
2
Physics 211R: Lab
Report
Template
Torque and Angular Momentum
PRINT THESE PAGES (p.
2

3
) WHEN YOU BEGIN AND INCLUDE IN YOUR REPORT
Q1. Back when we did linear & circular motion, we looked at what happened to the
velocity depending on the
direction of the acceleration (
a
= d
v
/dt). We’ll use those same
ideas in this lab for torque and angular momentum (
=
L
/dt).
For each box, say in a couple words what happens to the
v
or
L
vector and draw the new
v
or
L
vector for a
short
time later.
a
is parallel to
v
楳⁰慲a汬e氠l漠
L
a
is perpendicular to
v
楳⁰敲pe湤楣畬n爠瑯r
L
Q2. If a constant magnitude acceleration (or
F
net
) is always perpendicular to
v
, what kind
of
motion does the object undergo?
Q3. If a constant magnitude torque is always perpendicular to
L
, what kind of “motion”
does the angular momentum vector undergo?
Q4. If a force of (

50 N)
k
is exerted at a point (

0.2 m)
i
from the axis, what is the to
rque
on the system? (Express in terms of a unit vector.)
a
v
a
v
L
L
Penn State University
Physics 211R: Lab
–
Torque and Angular Momentum
3
Q5. If
L
is in the +
j
direction, what is happening to
L
(increase in magnitude, decrease in
magnitude, or being deflected) due to this torque?
Q6. In physics we use “right

handed” coordinate systems, defined as
. In the
figure below, you are given two of these coordinate directions. Draw in the missing unit
vector.
Q7.
Write an equation to determine the
Angular Velocity
D
of the Gyr
oscope’s Disk
given the Angular Velocity
S
of the Rotary Motion Sensor. (Are they connected at their
outer edges or do they share the same axle?) This equation is necessary for the rest of the
laboratory, so check with an instructor if you are not sure of your answer.
D
=
Q10. The precession of the gyroscope gives us a way to calculate the net torque on the
gyroscope using
= d
L
/dt. Note that we only care about rotation in the x

z plane since
the torque is always in that plane (why
can’t
the torque have a y

component?). So
instead
of
L
itself, we focus on
L
xz
=
L
sin
(where
=90
o
when the gyroscope rod is horizontal).
To see that relationship, consider the diagram below showing the change in angular
momentum for a
very short
time dt:
From trig
, what is d
楮⁴e牭猠潦
Ⱐ摴Ⱐ慮搠L
xz
?
(Hint: for very small angles tan
㴠
楮a摩慮猩
啳U⁴桡琠 灲p獳s潮⁴漠睲楴e⁴桥⁰ 物潤r
琠景t⁴桥 a湧畬慲
浯me湴畭⁶ c瑯爠瑯⁰牥ce獳y′
⡩渠
瑥t浳映
a湤nL
xz
)
What is the
precession
angular frequency
prec
? (in
terms of
a湤nL
xz
)
L
xz
at
t = 0
dL
xz
=
dt
d
L
xz
at
t =
dt
Penn State University
Physics 211R: Lab
–
Torque and Angular Momentum
4
Warm

Up: Understanding Torque and Angular Momentum (10 min max)
Answer all these questions (Q1

6
) on your printed sheet
Q1. Back when we did linear & circular motion, we looked
at what happened to the
velocity depending on the direction of the acceleration (
a
= d
v
/dt). We’ll use those same
ideas in this lab for torque and angular momentum (
=
L
/dt).
For each box
on your printed sheet
, say in a couple words what happens to the
v
or
L
vector and draw the new
v
or
L
vector for a
short
time later.
Q2. If a constant magnitude acceleration (or
F
net
) is always perpendicular to
v
, what kind
of motion does the object undergo?
Q3. If a constant magnitude torque is always perpendicular
to
L
, what kind of “motion”
does the angular momentum vector undergo?
Q4. If a force of (

50 N)
k
is exerted at a point (

0.2 m)
i
from the axis, what is the torque
on the system? (Express in terms of a unit vector.)
Q5. If
L
is in the
j
direction, what is happening to
L
(increase in magnitude, decrease in
magnitude, or being deflected) due to this torque?
Q6. In physics we use “right

handed” coordinate systems, defined as
. In the
figure on your
printed sheet
, you are given two of thes
e coordinate directions. Draw in
the missing unit vector.
Activity 1:
Understanding the Equipment
(15 minutes max)
In this activity, you are using the
Rotary Motion Sensor
. If you did the rotational
kinematics laboratory, you have already used this devi
ce, but may not have realized it (it
was attached to the spinning disc’s axle to measure
(t), and thus calculate
(t) and
(t).)
1.
Measure and record the
radius
of the gyroscope’s disk and record it in the table
below
. The radius of the Rotary Motion
Sensor’s disk has already been carefully
measured for you using a caliper.
Radius of Gyroscope’s Disk
Radius of Rotary Motion Sensor’s disk
2.759 cm
2.
Create a graphing window to display
Angular Velocity vs. Time
. Are you
measuring the angular
velocity of the sensor (
S
) or of
the gyroscope
disc (
D
)?
Are they the same?
Penn State University
Physics 211R: Lab
–
Torque and Angular Momentum
5
Q7.
Write an equation to determine the
Angular Velocity
D
of the Gyroscope’s Disk
given the Angular Velocity
S
of the Rotary Motion Sensor. (Are they connected at their
outer edges or do they share the same axle?) This equation is necessary for the rest of the
laboratory, so check with an instructor if you are not sure of your answer.
Write this
equation on your pr
inted sheet along with a brief justification.
Lab Activity
2
: Qualitative Observations of the Gyroscope
(15 min max)
Q8.
In which direction does the +Z axis point?
All answers in this lab should be
consistent with these coordinate axes
. Notice that the pivot point is located at the origin
in the above photo.
Q9.
From the viewpoint shown in the picture above, complete the following table by
clearly writing your observations and an explanation.
Use reference to the X,
Y
&
Z
axes
sh
own above
and the appropriate Right Hand Rule.
Penn State University
Physics 211R: Lab
–
Torque and Angular Momentum
6
Tasks
Questions
Observations / Explanations
(1

a) While the apparatus is at
rest, apply a
downward
(−Y
direction) force to the
left
end
of the rod.
Relative to the pivot, what is
the direction of the Torque
exerted
by the force?
(1

b) While the apparatus is at
rest, apply an
upward
(+Y
direction) force to the
left
end
of the rod.
Relative to the pivot, what is
the direction of the Torque
exerted
b
y the force?
(1

c) While the apparatus is at
rest, apply a
downward
(−Y
direction) force to the
right
end
of the rod.
Relative to the pivot, what is
the direction of the Torque
exerted
by the force?
(1

d) While the apparatus is at
rest, apply an
upward
(+Y
direction) force to the
right
end
of the rod.
Relative to the pivot, what is
the direction of the Torque
exerted
by the force?
(2

a) Rotate the disk so that the
top
edge of the disk is moving
in the
+Z
direction.
What is the direction of the
angular momentum of the disk?
What is the direction of the
angular velocity of the disk?
(2

b) Rotate the disk so that the
top
edge of the disk is moving
in the −
Z
direction.
What is the direction of the
angular momentum of the disk?
What is the
direction of the
angular velocity of the disk?
(3

a) While the disk is rotating
as in
(2

a)
, hang a mass from
the left end of the rod.
Describe what happens to the
motion of the gyroscope
relative to the pivot.
(3

b) While the disk is rotating
as in
(2

b)
, hang a mass from
the left end of the rod.
Describe what happens to the
motion of the gyroscope
relative to the pivot.
(4

a) Repeat the set up in (3

a):
spin the disk
quickly
and then
spin it
slowly
.
How does
the angular speed of
the disk affect the motion of
the gyroscope around the pivot
as observed in (3

a)?
(4

b) Repeat the set up in (3

b):
spin the disk
quickly
and then
spin it
slowly
.
How does the angular speed of
the disk affect the motion of
the
gyroscope around the pivot
as observed in (3

b)?
Penn State University
Physics 211R: Lab
–
Torque and Angular Momentum
7
Lab Activity
3
: Quantitative
Observations of the Gyroscope
(30 min)
Q10. In this activity, we will be measuring the precession frequency
prec
of the
gyroscope and comparing it to a theoretical calculation based on
net
= d
L
/dt
.
On your
printed sheet
, do the analysis described there to determine a theoretical relationship
between
prec
,
, and the angular momentum in the x

z plane, L
xz
.
Q11.
C
omplete the data table below and determine the magnitudes of the quantities in the
table. Record your results and clearly explain your methods and/or calculations in the
table below. Notice that some of the information has already been placed into the tab
le.
Also, be aware that there are two different ω’s
and two different angles (
and
)
.
Quantity
Result
(Magnitude)
Explanation of how your results
were obtained…
M = total mass hanging from lever
arm (kg)
F = weight of hanging mass (N)
r =
distance from pivot to hanging
mass (m)
= Angle between vectors F and r
(
)
= Torque (N m)
(
on rod
caused by hanging mass)
I = Rotational Inertia of Disk (kg
m
2
)
0.0136
This value was measured for you.
D
= Angular Velocity
of Disk
(rad/s)
L= Angular Momentum of Disk
(kg m
2
/s)
L
xz
= Angular momentum of Disk
in x

z plane (i.e., L sin
) (kg m
2
/s)
= Angular Displacement of
Rod (rad)
(about the Y axis)
(360
)
This value was chosen for you. If
you would like to use
another value,
simply make note of it.
t = time it takes the rod to rotate
(sec)
Use the Record and Stop buttons on
Data Studio™ to measure the time it
takes the gyroscope to rotate 360
around the Y axis.
prec
= angular velocity of
Rod
due to precession
(rad/s)
–
measured
prec
(rad/s)

predicted
from
and
L (Question 10)
Penn State University
Physics 211R: Lab
–
Torque and Angular Momentum
8
Q12. Use your results for
prec
to make a case whether the theoretical calculation in terms
of
a湤n
L
matches your experimental measurements.
Explain
your reasoning clearly and
completely. Defend your position by calculating a percent difference and interpreting the
meaning of it.
Lab Activity
4
: Conservation of Angular Momentum
(15 minutes)
Q13.
Using the appropriate Right Hand Rule,
complete the table below.
Initial Angular Momentum (kg m
2
/s)
of …
direction
The Bicycle Wheel
The Person (and stool)
Q14.
Quickly turn the wheel upside down so that its axis of rotation is again vertical. Record the
resulting motion in the table below.
Final Angular Momentum (kg m
2
/s)
of …
direction
The Bicycle Wheel
The Person (and stool)
Q15.
Assume that (1)
Angular Momentum is conserved and (2) the value of the Angular
Momentum of the Bicycle Wheel given below is correct. Complete the rest of the tables below,
using the directions observed above. Hint: A vector addition diagram is extremely helpful.
Initial Angular Momentum (kg m
2
/s) of …
magnitude
Direction
The Bicycle Wheel
0
.294
The Person (and stool)
The Total Momentum
Final Angular Momentum (kg m
2
/s) of …
magnitude
Direction
The Bicycle Wheel
The Person (and stool)
The Total Momentum
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