Rotation and Torque

Lecture 09

Thursday: 12 February 2004

ROTATION: DEFINITIONSROTATION: DEFINITIONSROTATION: DEFINITIONSROTATION: DEFINITIONS

• Angular position: • Angular displacement: 2 – 1 =

ttt

12

12ave :locity Angular ve Ave.

Instantaneous Angular velocity:

dt

d

•Use your right hand

•Curl your fingers in the direction of the rotation

•Out-stretched thumb points in the direction of the angular velocity

What is the direction of the angular velocity?

DEFINITIONS (CONTINUED)DEFINITIONS (CONTINUED)

dt

d

ttt

:onacceleratiangular ousInstantane

:onacceleratiangular Average

12

12avg

Direction of Angular Acceleration

The easiest way to get the direction of the angular acceleration is to determine the direction of the angular velocity and then…

• If the object is speeding up, velocity and acceleration must be in the same direction.

• If the object is slowing down, velocity and acceleration must be in opposite directions.

For constant For constant

221

00

0

tt

t

221

0

021

0

020

2

)(

)(2

tt

t

x v a

Relating Linear and Angular Variables

Relating Linear and Angular Variables

rr

r

r

va

ra

rv

rs

c

t

2222

Three Accelerations

1. Centripetal Acceleration (radial component of the linear acceleration)-always non-zero in circular motion.

2. Tangential Acceleration (component of linear acc. along the direction of the velocity)-non-zero if the object is speeding up or slowing down.

3. Angular Acceleration (rate of change in angular velocity)-non-zero is the object is speeding up or slowing down.

r

vac

2

raT

Energy Considerations

Although its linear velocity v is zero, the rapidly rotating blade of a table saw certainly has kinetic

energy due to that rotation.

How can we express the energy?

We need to treat the table saw (and any other rotating rigid body) as a collection of particles with different

linear speeds.

KINETIC ENERGY OF ROTATION

KINETIC ENERGY OF ROTATION

2

221

222122

21

221

221

Where ii

iiii

ii

iii

iii

rmI

IK

rmrmK

rv

vmKK

vmK

Defining Rotational Inertia

•The larger the mass, the smaller the acceleration produced by a given force.

•The rotational inertia I plays the equivalent role in rotational motion as mass m in translational motion.

amF

•I is a measure of how hard it is to get an object rotating. The larger I, the smaller the angular acceleration produced by a given force.

Determining the Rotational Inertia of an Object

1. For common shapes, rotational inertias are listed in tables. A simple version of which is in chapter 11 of your text book.

2. For collections of point masses, we can use :

where r is the distance from the axis (or point) of rotation.

3. For more complicated objects made up of objects from #1 or #2 above, we can use the fact that rotational inertia is a scalar and so just adds as mass would.

I is a function of both the mass and shape of the object. It also depends on the axis of rotation.

Ni

iiirmI

1

2

Comparison to TranslationComparison to Translation

• x • v • a • m I

• K=1/2mv21/2I2

Force and TorqueForce and Torque

I

Torque as a Cross Product

(Like F=Ma)The direction of the Torque is always in the direction of

the angular acceleration.

• For objects in equilibrium, =0 AND F=0

sinFr

Fr

Torque Corresponds to Force

Torque Corresponds to Force

• Just as Force produces translational acceleration (causes linear motion in an object starting at rest, for example)

• Torque produces rotational acceleration (cause a rotational motion in an object starting from rest, for example)

• The “cross” or “vector” product is another way to multiply vectors. Cross product results in a vector (e.g. Torque). Dot product (goes with cos ) results in a scalar (e.g. Work)

• r is the vector that starts at the point (or axis) of rotation and ends on the point at which the force is applied.

An Example

W

Forces on “extended” bodies can be viewed as acting on a point mass (with the same total mass)

At the object’s center of mass (balancing point)

xmg

SinFr

x

Determining Direction of A CROSS PRODUCT

Fr

Angular Momentum of a Particle

• Angular momentum of a particle about a point of rotation:

• This is similar to Torques

SinFr

Fr

SinPrl

prl

Find the direction of the angular momentum vector-Right hand

rule

P

P

r

r

Does an object have to be moving in a circle to have angular momentum?

• No.

• Once we define a point (or axis) of rotation (that is, a center), any object with a linear momentum that does not move directly through that point has an angular momentum defined relative to the chosen center as

p

prL