Chapter 10 - Rotation Definitions: –Angular Displacement –Angular Speed and Velocity –Angular Acceleration –Relation to linear quantities Rolling Motion Constant Angular Acceleration Torque –Vector directions –Moment Arm Newton’s 2nd Law for Rotation Calculating Rotational Inertia –Moment of inertia –Using the table –Parallel Axis Theorem –Perpendicular Axis Theorem Conservation of Angular Momentum Rotational Kinetic Energy Work and Rotational Kinetic Energy

Radius vs. position vector

Kinematics Memory Aid Forces cause acceleration!!!

Velocity Average velocity Instantaneous velocity

Angular Displacement

Angular Velocity Average angular velocity Instantaneous angular velocity

Acceleration Average acceleration Instantaneous acceleration

Angular Acceleration Average angular acceleration Instantaneous angular acceleration

Rotational Kinematics Memory Aid What causes angular acceleration?

Converting angular to linear quantities velocity tangential acceleration

Radial acceleration

Frequency vs. angular velocity Frequency –Cycles per time interval –Revolutions per time interval –Hertz Angular velocity –Radians per time interval –Sometimes called angular frequency –Radians/sec

Constant Acceleration

Constant Angular Acceleration

Problem 1 A record player is spinning at 33.3 rpm. How far does it turn in 2 seconds. The motor is shut off. The record player spins down in 20 seconds (assume constant deceleration). –What is the angular acceleration? –How far does it turn during this coast down?

Vector nature of angular quantities

Rolling without slipping

Problem 2 A cylinder of radius 12 cm starts from rest and rotates about its axis with a constant angular acceleration of 5.0 rad/s 2. At t = 3.0 sec, what is: –Its angular velocity –The linear speed of the point on the rim –The radial and tangential components of acceleration of a point on the rim.

Torque causes angular acceleration Torque is the moment of the force about an axis Product of a force and a lever arm Rotational Analog to Newton’s 2 nd Law

What if the force is not perpendicular?

Vector Multiplication – Cross Product

Right Hand Rule II

Vector Multiplication – Scalar Product

The Torque Vector R R

Problem 3 Find the net torque on the wheel about the center axle

Rotational Inertia Moment of inertia for a single particle

General Moment of Inertia

Problem 4 Three equal point masses are rotating about the origin at 2 rad/sec. The masses are located at (4m, 0) (0, 4m) and (4m, 4m). Each mass is 2 kg Find the moment of inertia.

Moment of inertia of a uniform cylinder

See Figure in book Moments of Inertia of various objects If particular axis is not in the table, use the parallel axis theorem:

Problem 5 A disk with radius, R, and mass, M, is free to rotate about its axis. A string is wrapped around its circumference with a block of mass, m, attached. This block is released from rest and falls. Find the tension in the string Find the acceleration Find the velocity after the mass has fallen a distance, h. m R M

Angular Momentum If there are no torques:

Two conservation of angular momentum demonstrations

Precession

Kepler’s 2nd Law The Law of Areas –A line that connects a planet to the sun sweeps out equal areas in equal times.

Rotational Kinetic Energy

Problem 6 - Energy A disk with radius, R, and mass, M, is free to rotate about its axis. A string is wrapped around its circumference with a block of mass, m, attached. This block is released from rest and falls. Find the tension in the string Find the acceleration Find the velocity after the mass has fallen a distance, h. m R M

Sphere rolling down a hill Find the velocity at the bottom of the hill? Mass = M, initially at rest

Which is fastest?