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PHY 133 Lab 7 ­ Angular Momentum [Stony Brook Physics Laboratory Manuals]
Stony Brook Physics Laboratory Manuals
PHY 133 Lab 7 - Angular Momentum
The purpose of this lab is to study the relationship between torque, moment of inertia, and angular acceleration, and to verify the
conservation of angular momentum.
Equipment
rotating table (with cylinder for wrapping string)
photogate attachment to rotating table
pulley and table clamp
disk with handle
mass with string (wraps around cylinder of rotating table)
vernier caliper (to measure the diameter of the cylinder)
interface box
computer
Introduction
In this experiment, we will investigate the “rotational analog” to linear motion, which is angular (or rotational) motion. Although a
previous lab covered uniform circular motion, in this lab, we will explore the angular forces (called “torques”) that cause angular
acceleration. Whereas linear quantities like distance, velocity, and acceleration are measured in meters, their angular counterparts
(angular distance (θ) , angular velocity (ω) , and angular acceleration (α)) may be calculated by dividing by the radius of the circular
path being traversed. Hence, you may recall the relations:
s = rθ v = rω a = rα
(1)
Similarly, we can develop a notion of an angular “distribution” of mass of a rotating object out to some radius r . This “angular
mass” is known as moment of inertia (I ). In the case of a point particle of mass m rotating about a central point at a radius r , the
moment of inertia of the particle is related to its mass via:
I = mr
2
(2)
Despite the simplicity of this expression for a point particle, the moment of inertia for an arbitrarily-shaped object may be much
more complex than equation (2). For most uniformly dense objects of all shapes (like rods, disks, and spheres), however, the
expression is generally proportional to mr 2 . You can find these expressions in any introductory physics textbook, or online.
Using this rotational framework, then, one might consider the angular analog of other important physical quantities, such as
momentum p and force F . Using the same conversion process as in equation (1), we may write expressions for the angular
momentum L and angular force (called “torque”) τ :
L = rp τ = rF
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For simplicity, we assume that the radial direction and the direction of motion are perpendicular, as in this experiment.
We may also obtain expressions for angular momentum and torque by taking the rotational analog of linear momentum and force:
p = mv ⟶ L = I ω
(4)
F = ma ⟶ τ = I α
(5)
Thus, using these relations, we can investigate, predict, and explain rotational motion. In this experiment in particular, we will explore
the rotational analog of two topics covered in previous labs: acceleration and conservation of momentum. Therefore, we will divide
the lab into two parts.
In Part I, we will measure the angular acceleration α of a rotating table due to an external torque, and obtain its moment of inertia I .
The equation which relates the net torque τnet to the moment of inertia and the net angular acceleration αnet of a rotating object is:
τnet = I α net
(6)
We can measure this angular acceleration by measuring the angular velocity ω as a function of time t . For a constant angular
acceleration, the equation which gives the dependence of ω on t is:
ω = ω0 + αt
(7)
Using this equation, we will find the net angular acceleration αnet and the angular acceleration due to friction alone αf r , which are
both needed to calculate the moment of inertia I of the rotating table.
In Part II, we will investigate the conservation of angular momentum L of the rotating table after a “collision” with another object.
By dropping a heavy disk on top of the spinning platform, and measuring the angular velocity before and after the drop, we will test
this conservation by expecting that:
′
′
′
L = L ⟶ I ω = I ω
(8)
where the un-primed quantities are the values of moment of inertia of the platform I and angular velocity ω before the collision, and
the primed quantities are the values of the total moment of inertia I ′ of the platform+disk combination and angular velocity ω′ after
the collision. To calculate the final moment of inertia, you simply add the moment of inertia of the platform alone and the moment
of inertia of the disk: I ′ = I + Idisk .
Measurement of the moment of inertia of the rotating table
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The apparatus used for this lab is sketched in the figure above. A small cylinder with string wound around it is attached to a rotating
platform. The tension T due to a weight of mass m attached to the string provides an external torque, τext tangent to the surface of
the cylinder. When the mass is released, the platform rotates, and a photogate registers the entry/exit times for the 4 black strips
along the plastic rim of the table. From the 90o angle (or π ≈ 1.571 radians) between the strips, the angular velocity ω is
2
computed. However, the system is not frictionless. As the mass falls and the string tension applies the external torque to the cylinder,
there is also friction along the cylinder axis that causes it to slow down, due to a frictional torque τf r . This frictional torque opposes
the external torque, thus causing a smaller net torque τnet = τext − τf r . This net torque gives the rotating platform an angular
acceleration αnet , which is measured from the rate of increase of the angular velocity ω. The frictional torque τf r causes a frictional
deceleration αf r .
The moment of inertia I of the rotating platform may be calculated using:
I =
mr (g − rαnet )
(9)
|αf r | + αnet
where g is the acceleration due to gravity and r is the radius of the cylinder beneath the platform around which you wind the string.
You should derive this equation in your lab report by considering that the net torque is τnet = τext − τf r = I αnet , the
frictional torque is τf r = I αf r , and the applied external torque is τext = T r = m(g − a)r. Use free-body diagrams to help
you visualize if necessary!
Preliminary steps
Use the vernier caliper to measure the diameter of the cylinder under the rotating platform. Before asking your lab
instructor/TA about how to do this, follow the simple instructions online either here
[http://www.technologystudent.com/equip1/vernier3.htm] or here [http://www.physics.smu.edu/~scalise/apparatus/caliper/]. Be sure
to apply the caliper with a close fit around the cylinder, and remove the caliper without changing the distance between its
“jaws.” Reading the caliper scale gives the diameter of the cylinder, from which you can find its radius r . Assume an
uncertainty of σ r = 0.5 mm, and record these values.
Make sure the string is long enough to reach from the small cylinder over the pulley and down to the floor. Then, hooking
the mass onto a loop at the bottom end, rotate the platform to wind the string around the base of the cylinder. This
maximizes the number of data points you will record before the mass m hits the floor.
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Connect the photogate output wire to the input of the interface box labeled “DIG/SONIC 1.” Turn on the computer, and
double-click the Desktop icon labeled “Exp_6_omega_t.” A “Sensor Confirmation” window should appear, and you should
make sure “Photogate” is selected, then click “Connect.” A LoggerPro window should appear, with a spreadsheet (having
columns labeled “Time,” “Velocity,” “Status 1”) on the left, and an empty graph of velocity vs. time on the right. Test the
photogate by blocking/unblocking its beam with your finger, and see that the red LED flashes on/off.
Input your value of the “characteristic distance” d into the LoggerPro program. For this experiment, this d value is the
angular distance (in radians) from one black strip to the next on the rim of the rotating platform. As mentioned before, this
π
angle is 2 ≈ 1.571 radians. To enter this value, under the “Data” tab at the top of the window, click “User Parameters.”
Input your value for d into the row labeled “PhotogateDistance1,” and change the “Places” and “Increment” values if
necessary. Then, click “OK.”
Part I: Computing the moment of inertia I of the platform
Measurement of αf r
For this part, make sure that the string is not attached to the cylinder (it can get tangled). Also, ensure that the photogate
beam is vertical, perpendicular to the rim of the rotating platform. Start the table rotating slowly, and click the green “Collect”
button at the top of the LoggerPro window. After the “Waiting for data…” text appears, the program should record data
values of (angular) velocity and time. Note: The SI unit for ω is radians/second, so ignore the units of linear velocity
(m/s) shown on the data table.
Let the table spin for 15-20 seconds, then click the red “STOP” button. (Note: If the data collection times out too early, go to
the “Experiment” tab, click “Data Collection,” and increase the duration of the trials as needed.) You should also see a
roughly linearly decreasing plot of (angular) velocity vs. time. If your plot is not straight, repeat the trial. Your data points may
waver up and down about a straight line, but this effect (due to a slight wobble in the rotating table) will be removed by linear
fitting.
Copy 10-15 pairs of angular velocity and time values into your notebook. To obtain an estimate of this angular deceleration
caused by friction, αf r , you can either use the Linear Fitting tool directly on your LoggerPro graph, or the web-based
Plotting Tool. However, do not take any uncertainty in αf r .
Measurement of αnet
Having checked that your string is long enough to reach the floor when unwound, attach a mass m = 200 g to the free end
of the string. Loop the string over the pulley, and position the photogate as in the procedure for finding αf r . Then, rotate
the platform to wind the string around the cylinder, raising the mass from the floor and up to the pulley.
When you're ready, click the green “Collect” button on LoggerPro, wait for the “Waiting for data…” text to appear, then
release the platform. It should begin rotating slowly, and speed up as the mass falls. Click the red “STOP” button once the
mass hits the floor. Make sure that ~15 pairs of velocity and time values appear on the LoggerPro spreadsheet, and copy
them into your notebook.
To obtain an estimate of this net angular acceleration αnet caused by the external torque AND the frictional torque
combined, you can again either use the Linear Fitting tool directly on your LoggerPro graph, or the web-based Plotting Tool.
And again, do not take any uncertainty in αnet .
Calculation of moment of inertia I
You have now measured everything necessary to calculate the moment of inertia I of the rotating platform using equation (9) above.
Despite the many input quantities, we will only consider the dominant uncertainty, which is the uncertainty in the radius, σ r . (This is
because g is much greater than rαnet , and the relative uncertainty in the mass, σ m , is very small.) Hence, assume that the relative
σ
σ
uncertainty in I equals the relative uncertainty in r , or:
=
. Record your values for I and σ I into your lab notebook.
r
I
r
I
Part II: Testing the conservation of angular momentum L
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Now that we've determined the moment of inertia of the rotating system, we will attempt to verify the principle of conservation of
angular momentum L. Similar to the notion of conservation of linear momentum p, we expect that a system with no applied
external torques on it will maintain a constant total angular momentum. Thus, if we simulate a “collision” between two rotating
objects such that no external torques act on them, the total angular momentum of the pair should remain constant. In this
experiment, one object will be the rotating platform (with no string and mass attached), and the other object will be a non-rotating
heavy metal disk dropped vertically onto the platform. Hence, we seek to prove equation (8) above for this case.
To do this, we will need the initial and final moments of inertia of the two objects, as well as the initial and final angular velocities of
the two objects. Before the “collision” occurs, the platform with moment of inertia I will be rotating with an angular velocity ω, and
the disk with moment of inertia Idisk will be held stationary without any angular rotation. Then, after the disk is dropped on top of
the platform, the platform+disk combination with total moment of inertia I ′ = I + Idisk will be rotating with an angular velocity
′
′
ω . Since we already have the value of I , the first step is to calculate the value of I (and its uncertainty σ I ).
′
For simplicity, treat the heavy metal disk as a solid, thin disk of mass M and radius R . Then, looking up the expression for moment
1
of inertia of such a uniformly dense object, we find that it may be expressed as: Idisk = M R2 . Record the mass value labeled
2
on the disk, and then measure its radius using a meter stick. Assume uncertainties in these quantities of σ M = 1 g and σ R = 1
mm. Then, compute the value of Idisk and its uncertainty σ I . Using these values, and the values for I and σ I from before,
calculate the final moment of inertia of the platform+disk combination I ′ = I + Idisk and its uncertainty σ I .
disk
′
Note: These equations only hold true if the axis of rotation of the platform+disk combination after the collision is the
same as the initial axis of rotation of the platform alone. Hence, it is essential that you drop the disk onto the rotating
platform so that their axes overlap!
Now, you're ready to collect some data:
Remove the string from the cylinder under the rotating platform. Prepare the LoggerPro program for data collection as in the
previous parts of the lab. When you're ready, carefully spin the rotating platform. (Make sure it does not wobble!) Click the
green “Collect” button, and collect data for ~5 seconds. You should notice a slight decrease in the velocity vs. time plot, due
to friction.
While the platform is spinning, hold the metal disk only slightly (~1 cm) above the rotating platform, and drop it so that the
rim of the disk overlaps with the rim of the platform. This means that their axes are overlapping. You will likely hear a loud
banging sound, so prepare yourself! Now, the plot of angular velocity vs. time on LoggerPro should look roughly like this:
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The data along the upper line correspond to the initial angular velocity ω, and the data along the lower line correspond to the
final angular velocity ω′ . Any data points in between, at the discontinuity (or, the “jump”) in the plot, may be disregarded.
From the spreadsheet values in the LoggerPro window, record the angular velocity values of 3 data points on either side of
the discontinuity in the plot. Note: The values must be as close to the discontinuity, and to each other, as possible!
Calculate the average value of each set of 3 points for ω and ω′ . For their uncertainties, σ ω and σ ω , use the formula
′
max−min
2
for each set of 3 values. Record all values in your lab notebook.
Finally, you have everything needed to determine whether angular momentum was conserved during this rotating “collision.” Begin
by calculating the initial and final values of angular momentum, L = I ω and L′ = I ′ ω′ . To find their uncertainties, σ L and σ L ,
use the uncertainty propagation relations with your values of σ I , σ ω , σ I , and σ ω from before. Then, verify (or dismiss) whether L
and L′ are the same within experimental uncertainty. Was angular momentum conserved in this experiment? If it was not, what are
possible reasons for this? In either case, what are some sources of error that may not have been fully accounted for in this lab? (For
example, would dropping the disk off-center from the axis of the rotating platform make a difference?)
′
′
′
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