Ch. 9 Rotational Ch. 9 Rotational KinematicsKinematics

AP Physics C

IntroductionIntroduction

Rotation of a rigid body about a fixed axis

A rigid body is not deformable; that is, the separations between particles remain constant.

Angular position, Angular position, θθ

Measured in radians

As a particle moves from A to B, it moves through an arc length of s.

s = rθ

What is a radian?

Conversions:Conversions:

1 revolution = _____ rad

1 revolution = _____ deg

Convert:1. 30o = ______ rad

2. 36 rad = _____ rev

3. 10 rev = _____ rad

4. 120o = _____ rev

Angular Displacement & Velocity:Angular Displacement & Velocity:

As the particle moves from A to B, its angular displacement, Δθ = θf - θi.

Its angular average velocity isIts instantaneous angular velocity is

avg t

0lim tt

d

dt

Angular Acceleration:Angular Acceleration:

If the instantaneous angular velocity of an object changes from ωi to ωf over a time interval of Δt, then the object has an average angular acceleration of

The instantaneous angular acceleration is

avg t

0limt

d

t dt

Sample Problem:Sample Problem:

The angular position θ, in radians, of a rotating object is given by the following equation:

a. Determine the object’s average angular speed from 1 s to 5 s.

b. Determine the object’s instantaneous angular speed as a function of t.

c. What is the object’s instantaneous speed at 3 s?d. What is the object’s average angular acceleration

from 1 s to 5 s?e. Determine the object’s instantaneous angular

acceleration as a function of time.f. What is the object’s instantaneous angular

acceleration at 3 s?

3 22 3t t t

Rotational Motion is analogous to Rotational Motion is analogous to linear motion.linear motion.

Linear Variables Rotational Variables

x

v

a

Constant Accelerated Motion:Constant Accelerated Motion:

Linear Motion Rotational Motion

Sample Problem:Sample Problem:

A wheel rotates with a constant angular acceleration of 2.5 rad/s/s. If the initial angular speed is 2.0 rad/s,

a.What is its final angular speed after 10.0 s?

b.What is its angular displacement ini. Radians?ii. Revolutions?iii. Degrees?

Relating Linear and Angular Relating Linear and Angular Kinematics:Kinematics:

What is the time derivative of s = rθ?

What is the tangential acceleration?

tan

ds dr

dt dtv r

tan

tan

dv dr

dt dta r

Centripetal Acceleration:Centripetal Acceleration:

Recall:

2

2

2

( )

c

c

c

va

r

ra

r

a r

Test Your Understanding:Test Your Understanding:

Information is stored on a CD or DVD in a coded pattern of tiny pits. The pits are arranged in a track that spirals outward toward the rim of the disc. As the disc spins inside a player, the track is scanned at a constant linear speed. How must the rotation speed of the disc change as the player’s scanning head moves over the track?

Sample Problem:Sample Problem:

A discuss thrower moves the discus in a circle of radius 80.0 cm. At a certain instant, the thrower is spinning at an angular speed of 10.0 rad/s and the angular speed is increasing at 50.0 rad/s/s. At this instant, find the tangential and centripetal acceleration of the discus and the magnitude of the acceleration.

Sample Problem:Sample Problem:

The smaller gear shown to the right has a radius of 5.0 cm and the larger one has a radius of 10.0 cm. If the angular speed of the smaller gear is 25 rad/s, what is the angular speed of the larger gear?

Rotational Kinetic Energy:Rotational Kinetic Energy:

Consider a rigid body that is made up of an infinite number of infinitesimal particles and rotating about a fixed axis, the kinetic energy of each particle is:

How would you write this kinetic energy expression in terms of angular speed?

21

2i iK mv

Rotational Kinetic EnergyRotational Kinetic Energy

What would be the total kinetic energy of the rigid body?

Moment of Inertia (MOI)Moment of Inertia (MOI)

MOI is a property of physics that indicates the relative difference in how easy or difficult it will be to set any object in motion about a defined axis of rotation.

MOI is always measured relative to a point of reference.

MOI depends on an object’s mass and on its shape.

MOI depends on the distribution of mass.

Moment of Inertia (MOI)Moment of Inertia (MOI)

Newton’s first law of motion states, “A body maintains the current state of motion unless acted upon by an external force.”

In linear motion, the measure of inertia refers to the mass of the system.

In rotational motion, the measure of inertia refers to the moment of inertia of the system.

Moment of Inertia (MOI)Moment of Inertia (MOI)

MOI for a system of particles:

Determine MOI of this system of 4 points masses, as they rotate about the:◦X-axis◦Y-axis◦Axis perpendicular to

the origin

2i iI m r

Moment of Inertia (MOI)Moment of Inertia (MOI)

MOI for a rigid body:

Pg. 342

2I r dm

Rotational Kinetic EnergyRotational Kinetic Energy

What is the total kinetic energy of a rigid body?

What is the MOI for a system of particles?

What is the kinetic energy of a rotating rigid body in terms of its MOI?

Sample Problem:Sample Problem:

Given the equilateral triangle to the right with equal masses of m at each vertex, determine the:◦MOI about an axis

perpendicular to the plane at the x and

◦The kinetic energy of the system about this axis.

Gravitational Potential EnergyGravitational Potential Energy

What is the gravitational potential energy for each particle of a rigid body?

What is the total gravitational potential energy of a rigid body?

Sample ProblemSample Problem

Determine the speed of the disk and the hoop, shown to the right, when they reach the bottom of the ramp. Let h represent the height that their CMs start at.

Sample ProblemSample Problem

The mass of the pulley is 2.5 kg and its radius is 0.25 m. If the mass on the right is 6.0 kg and the one on the left is 4.5 kg, what is the speed at which the 6.0 kg block hits the floor if it moved a distance of 1.0 m?

Sample ProblemSample Problem

What is the speed of the block when it has traveled a distance of h?

What is the angular speed of the pulley when the block has traveled a distance of h?

Parallel-Axis TheoremParallel-Axis Theorem

Determine the MOI of the rigid body about an axis that is perpendicular to the origin.

Sample ProblemSample Problem

Use the Parallel-Axis Theorem to determine the MOI about an axis that is perpendicular to point O.

Sample ProblemSample Problem

Use the Parallel-Axis Theorem, to determine the MOI about an axis that is parallel to the z-axis an a distance of r/2 from the z-axis.

Moment of Inertia CalculationMoment of Inertia Calculation

Calculate the MOI of a uniform thin rod with an axis of rotation that is perpendicular to its length.

Moment of Inertia CalculationMoment of Inertia Calculation

Calculate the MOI of a uniform hollow or uniform solid cylinder that is rotating about its axis of symmetry.