Chapter 9

Rotational Dynamics

9.1 The Action of Forces and Torques on Rigid Objects

In pure translational motion, all points on anobject travel on parallel paths.

The most general motion is a combination oftranslation and rotation.

9.1 The Action of Forces and Torques on Rigid Objects

According to Newton’s second law, a net force causes anobject to have an acceleration.

What causes an object to have an angular acceleration?

TORQUE

9.1 The Action of Forces and Torques on Rigid Objects

The amount of torque depends on where and in what direction the force is applied, as well as the location of the axis of rotation.

9.1 The Action of Forces and Torques on Rigid Objects

DEFINITION OF TORQUE

Magnitude of Torque = (Magnitude of the force) x (Lever arm)

FDirection: The torque is positive when the force tends to produce a counterclockwise rotation about the axis.

SI Unit of Torque: newton x meter (N·m)

9.2 Rigid Objects in Equilibrium

If a rigid body is in equilibrium, neither its translational motion nor its rotational motion changes.

0 xF 0yF 0

0 yx aa 0

9.2 Rigid Objects in Equilibrium

EQUILIBRIUM OF A RIGID BODY

A rigid body is in equilibrium if it has zero translationalacceleration and zero angular acceleration. In equilibrium,the sum of the externally applied forces is zero, and thesum of the externally applied torques is zero.

0 0yF0 xF

9.2 Rigid Objects in Equilibrium

Example 3 A Diving Board

A woman whose weight is 530 N is poised at the right end of a diving boardwith length 3.90 m. The board hasnegligible weight and is supported bya fulcrum 1.40 m away from the leftend.

Find the forces that the bolt and the fulcrum exert on the board.

9.2 Rigid Objects in Equilibrium

022 WWF

N 1480

m 1.40

m 90.3N 5302 F

22

WWF

9.2 Rigid Objects in Equilibrium

021 WFFFy

0N 530N 14801 F

N 9501 F

9.3 Center of Gravity

DEFINITION OF CENTER OF GRAVITY

The center of gravity of a rigid body is the point at whichits weight can be considered to act when the torque dueto the weight is being calculated.

9.3 Center of Gravity

21

2211

WW

xWxWxcg

9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis

TT maF

raT rFT

2mr

Moment of Inertia, I

9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis

2

2222

2111

NNN rm

rm

rm

2mr

Net externaltorque Moment of

inertia

9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis

ROTATIONAL ANALOG OF NEWTON’S SECOND LAW FORA RIGID BODY ROTATING ABOUT A FIXED AXIS

onaccelerati

Angular

inertia

ofMoment torqueexternalNet

I

2mrIRequirement: Angular acceleration must be expressed in radians/s2.

9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis

Example 9 The Moment of Inertia Depends on Wherethe Axis Is.

Two particles each have mass and are fixed at theends of a thin rigid rod. The length of the rod is L.Find the moment of inertia when this object rotates relative to an axis that is perpendicular to the rod at(a) one end and (b) the center.

9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis

(a) 22222

211

2 0 LmmrmrmmrI

Lrr 21 0mmm 21

2mLI

9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis

(b) 22222

211

2 22 LmLmrmrmmrI

22 21 LrLr mmm 21

221 mLI

9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis

9.5 Rotational Work and Energy

FrFsW

rs

Fr

W

9.5 Rotational Work and Energy

DEFINITION OF ROTATIONAL WORK

The rotational work done by a constant torque in turning an object through an angle is

RW

Requirement: The angle mustbe expressed in radians.

SI Unit of Rotational Work: joule (J)

9.5 Rotational Work and Energy

22122

2122

21 ImrmrKE

22212

21 mrmvKE T

rvT

9.5 Rotational Work and Energy

221 IKER

DEFINITION OF ROTATIONAL KINETIC ENERGY

The rotational kinetic energy of a rigid rotating object is

Requirement: The angular speed mustbe expressed in rad/s.

SI Unit of Rotational Kinetic Energy: joule (J)

9.6 Angular Momentum

DEFINITION OF ANGULAR MOMENTUM

The angular momentum L of a body rotating about a fixed axis is the product of the body’s moment of inertia and its angular velocity with respect to thataxis:

IL

Requirement: The angular speed mustbe expressed in rad/s.

SI Unit of Angular Momentum: kg·m2/s

9.6 Angular Momentum

PRINCIPLE OF CONSERVATION OF ANGULAR MOMENTUM

The angular momentum of a system remains constant (is conserved) if the net external torque acting on the system is zero.

9.6 Angular Momentum

Conceptual Example 14 A Spinning Skater

An ice skater is spinning with botharms and a leg outstretched. Shepulls her arms and leg inward andher spinning motion changesdramatically.

Use the principle of conservationof angular momentum to explainhow and why her spinning motionchanges.

Problems to be solved

• 9.6, 9.12, 9.14, 9.21, 9.25, 9.40, 9.51, 9.61, 9.69, 9.74

• B9.1 A bicycle wheel has a mass of 2kg and a radius of 0.35m. What is its moment of inertia? Ans: 0.245kgm2

• B9.2 A grinding wheel, a disk of uniform thickness, has a radius of 0.08m and a mass of 2kg. (a) What is its moment of inertia? (b) How large a torque is needed to accelerate it from rest to 120rad/s in 8s? Ans: (a) 0.0064kgm2 (b) 0.096Nm

• B9.3 A student holding a rod by the centre subjects it to a torque of 1.4Nm about an axis perpendicular to the rod, turning it through 1.3rad in 0.75s. When the student holds the rod by the end applies the same torque to the rod, through how many radians will the rod turn in 1.0s?

Ans: 0.58rad

• B9.4 An ice skater starts spinning at a rate of 1.5rev/s with arms extended. She then pulls her arms in close to her body, resulting in a decrease of her moment of inertia to three quarters of the initial value. What is the skater’s final angular velocity? Ans: 2rev/s