Torque and Angular Momentum

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Torque and Angular Momentum
Experiment 13
Objective: To investigate the change in the angular momentum of an object due to an
applied torque.
DISCUSSION:
 = I L
Angular velocity  and angular momentum L=I
are vectors. The direction of these vectors are
given by the right hand rule.
The equations describing Newton’s 2nd
law for rotational motion are fully analogous to
the equations for linear motion. A net torque on
an object will change its angular velocity :
  I ,
where I is the moment of inertia, and the angular
acceleration  = d/dt. We may also write the 2nd Law to relate the net torque to angular
momentum L.
dL
,
 
r
dt
Thus, the direction of the applied torque is always in the
same direction as the resulting angular acceleration and
the change in the angular momentum.
Consider Fig. 1. A torque is applied on a rotating
disk by a hanging weight. The net torque on the system
is

T
T
  I
r   mg    I disk  I weight  
rmg   I disk  mr 2  
mg
Fig. 1
1
EXERCISE 1: Torque and Angular Acceleration
1. Setup the signal interface
a. Verify that the unit is plugged in and powered.
b. Plug the phone jacks from the motion detector into plugs 1 and 2 of the
interface
c. Connect your computer via the USB cable.
2. Setup DataStudio
a. When prompted, choose ‘Create Experiment’
b. Click ‘Add Sensor or Instrument’
c. Choose from ‘Science Workshop Digital Sensors’
d. From the list, select ‘Rotary Motion Sensor’
e. On the list of measurements, check the following
Angular Position, Ch 1&2
Angular Velocity, Ch 1&2
f. From the list of Displays, drag Graph to Data list above and drop it onto
Angular Position.
g. Drag Graph1 and drop it onto Angular Velocity.
3. Assemble the apparatus
a. Securely, attach the weights to each end of the bar. Then attach the bar to the
axel. Make sure it is balanced.
b. Using the string provided, measure the length between the two knots tied on
the string. Then wind the length of string between the two knots around
the larger of the two pulleys on the apparatus.
4. Perform the experiments
a. Attach a 20g mass to the end of the string. In DataStudio, click Start and the
drop the mass. The angular position and velocity will be recorded by the
computer
b. Wind the string around the smaller pulley and repeat the experiment.
c. Move each mass about halfway between the middle and end of the bar. Repeat
the experiment for each pulley.
DATA ANALYSIS
1. Finding the Moment of Inertia
a. Using the measured values of the acceleration from each experiment, calculate
the moments of inertia of the apparatus using the equations above.
b. Now, calculate the moment of inertia from the geometry of the apparatus for
each of the two configurations. .
c. Find the percent deviation between these values.
2
2. Finding the Torque
a. From your experimental value of the moment of inertia and the angular
acceleration, calculate the net torque on the apparatus. (Note, this is less
than the net torque on the apparatus plus the hanging mass.)
b. From the experimental data gathered after the hanging mass fell free, calculate
the torque of friction on the apparatus. Is this sufficient to explain the
experimental deviation found in the moment of inertia measurements?
EXERCISE 2: Gyroscopic Precession
A gyroscope which is not spinning and
not perfectly vertical will fall over. This is
due to the torque of the gyroscope’s own
weight.
A spinning gyroscope which is not
perfectly vertical will undergo a precession
motion. This is due to the torque of the
gyroscope’s own weight.
In both cases, the change in the in the
dL
gyroscope’s angular momentum
is in the
dt
same direction of the gravitational torque.
1. Observe a non-spinning gyroscope fall
over. What is the direction of the torque?
L
o
mg
2. With a quick pull of the supplied string, give the gyroscope a large angular momentum
in the red direction. Set the gyroscope on the table with the red end upward and observe
the precession.
What is the direction of the torque?
What is the direction of the precession? (Clockwise or counter-clockwise.)
Sketch a diagram of the top showing the direction of the angular momentum,
gravitational torque, and precession.
3. With the gyroscope still spinning in the red direction, flip it over so the red end is on
the bottom.
What is the direction of the torque?
What is the direction of the precession? (Clockwise or counter-clockwise.)
Again, sketch a diagram..
3
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