9 Rotation Rotational Kinematics: Angular Velocity and Angular Acceleration Rotational Kinetic Energy Calculating the Moment of Inertia Newton’s Second Law for Rotation Applications of Newton’s Second Law for Rotation Rolling Objects Hk: 29, 37, 41, 47, 67, 71, 79, 85, 105.

Rotation

Angular Velocity & Acceleration

Ex. Angular Velocity & Acceleration

Linear/Angular Relation for Rotation

Angular Kinematics

Rotational Kinetic Energy

Ex. Rotational KE

Continuous Objects

Parallel Axis Theorem

Ex. Parallel Axis Theorem

Tangential Acceleration

Newton’s Second Law for Rotation

Ex. Newton’s 2 nd Law for Rotation

Rolling Motion

Rotational Power

Summary Angular Velocity, Acceleration Rotational Kinetic Energy Calculating the Moment of Inertia Newton’s Second Law for Rotation Rotational Power Rolling Objects

Problems

09-1. A compact disk rotates from rest to 400 rev/min in 4.0 seconds. a) Convert 400 rev/min into radians/second. b) Calculate the average angular acceleration in rad/s/s. c) Calculate the number of radians and the number of revolutions subtended by the disk during this interval. d) How far does a point on the edge of the disk, radius 6.0cm, travel during this interval? e) If the disk has mass of 15.5 grams, what is its kinetic energy when rotating at 400 rev/min? f) What average power is needed to accelerate the disk to 400rev/min in 4.0 seconds? (this turns out to be a relatively small part of the total power consumed by a CD player)

a) Convert 400 rev/min into radians/second. b) Calculate the average angular acceleration in rad/s/s. c) Calculate the number of radians and the number of revolutions subtended by the disk during this interval.

d) How far does a point on the edge of the disk, radius 6.0cm, travel during this interval? e) If the disk has mass of 15.5 grams, what is its kinetic energy when rotating at 400 rev/min? f) What average power is needed to accelerate the disk to 400rev/min in 4.0 seconds? (this turns out to be a relatively small part of the total power consumed by a CD player)

09-2. System = 85 gram meter-stick & mass 65 grams at end. a) Rod is vertical with 65 gram end up. What is system PE-g wrt to the axis? b) System PE-g at bottom of swing? c) Calculate the moment of inertia of the system about the axis of rotation in SI units. d) If 90% of the gravitational potential energy lost in the swing of part (b) is converted to kinetic energy of the mass system, what is the rotational rate of the mass system in rad/s at the bottom of its swing? e) What is the speed of the 65 gram point mass at swing bottom?

a) PE-g wrt to the axis? b) System PE-g at bottom of swing? c) Moment of inertia?

d) 90% PE-g is converted to kinetic energy. What is the rotational rate of the system at the bottom of its swing? e) Speed of the 65 gram point mass at swing bottom?

09-3. Thin-walled circular cylinder mass 160 kg, radius 0.34 m. (a) Calculate the moment of inertia of the rotor about its own axis. (b) Calculate the rotational kinetic energy stored in the rotating rotor when it spins at rev/min. c) A 1400kg car slows from a speed of 20 m/s to 10 m/s and the rotational speed of the flywheel is increased from rpm to rpm. What percentage of the car’s kinetic energy change is funneled into the flywheel? d) If the flywheel above were accelerated in 10s, what minimum force applied at the edge of the disk would be required?

(a) Calculate the moment of inertia of the rotor about its own axis. (b) Calculate the rotational kinetic energy stored in the rotating rotor when it spins at rev/min.

c) A 1400kg car slows from a speed of 20 m/s to 10 m/s and the rotational speed of the flywheel is increased from rpm to rpm. What percentage of the car’s kinetic energy change is funneled into the flywheel?

d) If the flywheel above were accelerated in 10s, what minimum force applied at the edge of the disk would be required?

Ex. m1=m2=m3=m4 = m r1=r2=r3=r4 = a

Ex. m1=m2=m3=m4 = m r1=r2 = 0 r3=r4 = 2a

Example A car wheel angularly accelerates uniformly from 1.5rad/s with rate 3.0rad/s 2 for 5.0s. What is the final angular velocity? What angle is subtended during this time?

Ex: Changing Units