1. Two-Dimensional Rotational Dynamics
8.01
W10D2 Fall, 2006

2. Main Idea: Fixed Axis Rotation of Rigid Body
Torque produces angular acceleration about center of mass
is the moment of inertial about the center of mass
is the angular acceleration about center of mass

3. Torque as a Vector Force exerted at a point P on a rigid body.
Vector from a point S to the point P. P  S
Torque about point S due to the force exerted at point P:

4. Summary: Cross Product
Magnitude: equal to the area of the parallelogram defined by the two vectors
Direction: determined by
the Right-Hand-Rule

5. Properties of Cross Products

6. Cross Product of Unit Vectors
Unit vectors in Cartesian coordinates

7. Components of Cross Product

8. Concept Question: Cross Product
Consider two vectors with x > 0 and
with Fx > 0 and Fz > 0 . The cross product
points in the
+ x-direction
-x-direction
+y-direction
-y-direction
+z-direction
-z-direction
None of the above directions

9. Recall from Last Class: Rotational Kinematics
Individual element of mass
Radius of orbit
Tangential velocity
Tangential acceleration
Radial Acceleration

10. Dynamics: Newton’s Second Law and Torque about S
Tangential force on mass element produces torque
Newton’s Second Law
Torque about S
z-component of torque about S

11. Moment of Inertia and Torque
Component of the total torque about an axis passing through S is the sum over all elements
Recall: Moment of Inertia about and axis passing through S :
Summary:

12. Concept Question: Chrome Inertial Wheel
A fixed torque rotates is applied to the shaft of the chrome inertial wheel. If the four weights on the arms of the are slid out, the component of the angular acceleration along the shaft direction will
increase.
decrease.
remain the same.

13. Demo: Chrome Inertial Wheel
For a fixed torque: vary the moment of inertia will vary the angular acceleration.

14. Demo: Moment of Inertia Wheel
Measuring the moment of inertia.

15. Analysis: Measuring Moment of Inertia
Free body force diagrams and force equations:
Rotational equation:
Constraint:
Solve for moment of inertia:
Time to travel distance s:

16. Properties of Wheel Radius of disc: Mass of disc:
Mass of weight holder:
Theoretical result:

17. Group Problem: Atwood’s Machine
A pulley of mass mp, radius R, and moment of inertia Icm about the center of mass, is suspended from a ceiling. An inextensible string of negligible mass is wrapped around the pulley and attached on one end to an object of mass m1 and on the other end to an object of mass m2 , with
m1 > m2 . At time t = 0, the objects are released from rest. Find the magnitude of the acceleration of the objects.

18. Analysis: Atwood’s Machine
Free body force diagrams and force equations:

19. Analysis: Atwood’s Machine
Rotational motion equation:
Force equations:
Constraint condition:
Acceleration:
Time to
travel distance d:

20. Experiment 05: Moment of Inertia

21. Apparatus Connect output of phototransistor to channel A of 750.
Connect output of tachometer generator to channel B of 750.
Connect power supply.
Red button is pressed: Power is applied to motor.
Red button is released: Rotor coasts: Read output voltage using LabVIEW program.
Use black sticker or tape on white plastic rotor for generator calibration.

22. Calibrate Tachometer-generator
Spin motor up to full speed, let it coast. Measure and plot voltages for 0.25 s period. Sample Rate: 5000 Hz.
Count rotation periods to measure ω.
Program calculates average output voltage, angular velocity, and the calibration factor angular velocity per volt periods.

23. Rotor Moment of Inertia
Plot only the generator voltage for rest of experiment.
Use a 55 gm weight to accelerate the rotor.

24. Analysis: moment of inertia
Force and rotational equations while weight is descending:
Constraint:
Rotational equation while slowing down
Solve for moment of inertia:
Speeding up
Slowing down

25. Understand graph output to measure IR
The angular frequency along line A-B is increasing because
the weight has hit the floor and is tension in the string is no longer applying a torque.
the weight is descending and the tension in the string is applying a torque.
3. for some other reason.

26. Understand graph output to measure IR
The slope of the line B-C is equal to
the angular acceleration after the weight has hit the floor.
angular acceleration before the weight has hit the floor.
3. Neither of the above.

27. Measure IR: results Measure and record α1 and α2.
For your report, calculate IR: