m08: rolling, torque, and angular momentum

October 9, 2023

11-1 Rolling as Translation and Rotation Combined (1 of 6)

Learning Objectives

11.01 Identify that smooth rolling can be considered as a combination of pure translation and pure rotation
11.02 Apply the relationship between the center-of-mass speed and the angular speed of a body in smooth rolling

11-2 Forces and Kinetic Energy of Rolling (1 of 12)

Learning Objectives

11.03 Calculate the kinetic energy of a body in smooth rolling as the sum of the translational kinetic energy of the center of mass and the rotational kinetic energy around the center of mass.
11.04 Apply the relationship between the work done on a smoothly rolling object and its kinetic energy change.
11.05 For smooth rolling (and thus no sliding), conserve mechanical energy to relate initial energy values to the values at a later point.

11-2 Forces and Kinetic Energy of Rolling (2 of 12)

11.06 Draw a free-body diagram of an accelerating body that is smoothly rolling on a horizontal surface or up or down on a ramp.
11.07 Apply the relationship between the center-of-mass acceleration and the angular acceleration.
11.08 For smooth rolling up or down a ramp, apply the relationship between the object’s acceleration, its rotational inertia, and the angle of the ramp.

11-3 The Yo-Yo (1 of 3)

Learning Objectives

11.09 Draw a free-body diagram of a yo-yo moving up or down its string.
11.10 Identify that a yo-yo is effectively an object that rolls smoothly up or down a ramp with an incline angle of 90°.
11.11 For a yo-yo moving up or down its string, apply the relationship between the yo-yo’s acceleration and its rotational inertia.
11.12 Determine the tension in a yo-yo’s string as the yo-yo moves up or down the string.

11-4 Torque Revisited (1 of 7)

Learning Objectives

11.13 Identify that torque is a vector quantity.
11.14 Identify that the point about which a torque is calculated must always be specified.
11.15 Calculate the torque due to a force on a particle by taking the cross product of the particle’s position vector and the force vector, in either unit-vector notation or magnitude-angle notation.
11.16 Use the right-hand rule for cross products to find the direction of a torque vector.

11-5 Angular Momentum (1 of 6)

Learning Objectives

11.17 Identify that angular momentum is a vector quantity.
11.18 Identify that the fixed point about which an angular momentum is calculated must always be specified.
11.19 Calculate the angular momentum of a particle by taking the cross product of the particle’s position vector and its momentum vector, in either unit-vector notation or magnitude-angle notation.
11.20 Use the right-hand rule for cross products to find the direction of an angular momentum vector.

11-6 Newton’s Second Law in Angular Form (1 of 6)

Learning Objectives

11.21 Apply Newton’s second law in angular form to relate the torque acting on a particle to the resulting rate of change of the particle’s angular momentum, all relative to a specified point.

11-7 Angular Momentum of a Rigid Body (1 of 6)

Learning Objectives

11.22 For a system of particles, apply Newton’s second law in angular form to relate the net torque acting on the system to the rate of the resulting change in the system’s angular momentum.
11.23 Apply the relationship between the angular momentum of a rigid body rotating around a fixed axis and the body’s rotational inertia and angular speed around that axis.
11.24 If two rigid bodies rotate about the same axis, calculate their total angular momentum.

11-8 Conservation of Angular Momentum (1 of 7)

Learning Objectives

11.25 When no external net torque acts on a system along a specified axis, apply the conservation of angular momentum to relate the initial angular momentum value along that axis to the value at a later instant

11-9 Precession of a Gyroscope (1 of 7)

Learning Objectives

11.26 Identify that the gravitational force acting on a spinning gyroscope causes the spin angular momentum vector (and thus the gyroscope) to rotate about the vertical axis in a motion called precession.
11.27 Calculate the precession rate of a gyroscope.
11.28 Identify that a gyroscope’s precession rate is independent of the gyroscope’s mass.

11 Summary (1 of 3)

Rolling Bodies.
Torque as a vector.
Angular Momentum of a Particle.
Newton’s Second Law in Angular Form.
Angular Momentum of a System of Particles.
Angular Momentum of a Rigid Body.
Conservation of Angular Momentum.
Precession of a Gyroscope.

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