Unit 5: Rotations

Big Ideas:
1. Translational and rotational mechanics are described with corollary parameters.
2. Torque causes rotational acceleration.
3. The rotational inertia of an object is a measure of its resistance to change in its rotational state of motion.

Learner Objectives (as published by the College Board):
1. Students should understand the concept of torque, so they can:
A. Calculate the magnitude and direction of the torque associated with a given force.
B. Calculate the torque on a rigid object due to gravity.

2. Students should be able to analyze problems in statics, so they can:
A. State the conditions for translational and rotational equilibrium of a rigid object.
B. Apply these conditions in analyzing the equilibrium of a rigid object under the combined influence of a number of coplanar forces applied at different locations.

3. Students should develop a qualitative understanding of rotational inertia, so they can:
A. Determine by inspection which of a set of symmetrical objects of equal mass has the greatest rotational inertia.
B. Determine by what factor an object's rotational inertia changes if all its dimensions are increased by the same factor.

4. Students should develop skill in computing rotational inertia so they can find the rotational inertia of:
A. A collection of point masses lying in a plane about an axis perpendicular to the plane.
B. A thin rod of uniform density, about an arbitrary axis perpendicular to the rod.
C. A thin cylindrical shell about its axis, or an object that may be viewed as being made up of coaxial shells.

5. Students should be able to state and apply the parallel-axis theorem.

6. Students should understand the analogy between translational and rotational kinematics so they can write and apply relations among the angular acceleration, angular velocity, and angular displacement of an object that rotates about a fixed axis with constant angular acceleration.

7. Students should be able to use the right-hand rule to associate an angular velocity vector with a rotating object.

8. Students should understand the dynamics of fixed-axis rotation, so they can:
A. Describe in detail the analogy between fixed-axis rotation and straight-line translation.
B. Determine the angular acceleration with which a rigid object is accelerated about a fixed axis when subjected to a specified external torque or force.
C. Determine the radial and tangential acceleration of a point on a rigid object.
D. Apply conservation of energy to problems of fixed-axis rotation.
E. Analyze problems involving strings and massive pulleys.

9. Students should understand the motion of a rigid object along a surface, so they can:
A. Write down, justify, and apply the relation between linear and angular velocity, or between linear and angular acceleration, for an object of circular cross-section that rolls without slipping along a fixed plane, and determine the velocity and acceleration of an arbitrary point on such an object.
B. Apply the equations of translational and rotational motion simultaneously in analyzing rolling with slipping.
C. Calculate the total kinetic energy of an object that is undergoing both translational and rotational motion, and apply energy conservation in analyzing such motion.

10. Students should be able to use the vector product and the right-hand rule, so they can:
A. Calculate the torque of a specified force about an arbitrary origin.
B. Calculate the angular momentum vector for a moving particle.
C. Calculate the angular momentum vector for a rotating rigid object in simple cases where this vector lies parallel to the angular velocity vector.

11. Students should understand angular momentum conservation, so they can:
A. Recognize the conditions under which the law of conservation is applicable and relate this law to one- and two-particle systems such as satellite orbits.
B. State the relation between net external torque and angular momentum, and identify situations in which angular momentum is conserved.
C. Analyze problems in which the moment of inertia of an object is changed as it rotates freely about a fixed axis.
D. Analyze a collision between a moving particle and a rigid object that can rotate about a fixed axis or about its center of mass.