Chapter 8Chapter 8

Rotational MotionRotational Motion

Rotational MotionRotational Motion

Angular Distance ()o Replaces distance for rotational motiono Measured in

Degrees

Radians

Revolutions

Radian MeasureRadian Measure

r

r r

1 rad

1 rad = 57.3 degrees

2 rad in one circle

Windows CalculatorWindows Calculator

Rotational MotionRotational Motion

Speed of Rotation () = Angle covered/Time required

o = to Note similarity to v = x/t

o Measured in degrees/second radians/second revolutions/second

Rotational MotionRotational Motion

Angular Acceleration - Measures how angular velocity is changing () = /t Note similarity to a = v/to Measured in …

degrees/s2

radians/s2

revolutions/s2

Rotational InertiaRotational Inertia

Property of an object that resists changes in rotation• For linear motion mass was a measure of

inertia

• For rotational motion Moment of Inertia (I) is the measure of rotational Inertia

Moments of InertiaMoments of Inertia

Depends on …

o Mass of the Object

o Axis of Rotation

o Distribution of Mass in the Object

Moments of InertiaMoments of Inertia

Standard Shapes

Moment of InertiaMoment of Inertia

Inertia Bars Ring and Disk on Incline Metronome People walking Weighted Stick - Bare Stick

TorqueTorque

Product of Force and Lever Armo Torque = Force X Lever Arm

Examples:o Balanceo See-Sawo Wrench

W1d1 = W2d2

Sample Torque ProblemSample Torque Problem

(0.5 kg)(9.8 m/s(0.5 kg)(9.8 m/s22)(0.1 m) = (0.2 kg)(9.8 m/s)(0.1 m) = (0.2 kg)(9.8 m/s22)d)d

m 0.25 d

m) (0.1 kg 0.2kg 0.5

d

Line of Action

Lever Arm

F

Torque ExamplesTorque Examples

TorqueTorque

Just as unbalanced forces produce acceleration, unbalanced torques produce angular acceleration.

Compare:

F = ma = I

Center of MassCenter of Mass

Average position of the mass of an object

o Newton showed that all of the mass of the object acts as if it is located here.

o Find cm of Texas/USA

Finding the Center of MassFinding the Center of Mass

weight

Line of actionPivot point

Lever armTorque

No Torque

High JumperHigh Jumper

StabilityStability

In order to balance forces and torques, the center of mass must always be along the vertical line through the base of support.

Demo• Coke bottle• Chair pick-up

StabilityStability

Base of Support

StabilityStability

Which object is most stable?

Centripetal ForceCentripetal Force

Any force that causes an object to move in a circle.

Examples:• Carousel• Water in a bucket• Moon and Earth• Coin and hanger• Spin cycle

Centripetal forceCentripetal force

F = mac

= mv2/r

= mr2

Centrifugal forceCentrifugal force

Fictitious center fleeing forceo Felt by object in an accelerated reference

frame Examples:

o Car on a circular patho Can on a string

Space Habitat(simulated gravity)

r

Space Habitat(simulated gravity)

“Down” is away from the center The amount of “gravity” depends on

how far from the center you are.

Angular MomentumAngular Momentum

L = (rotational inertia) X (angular velocity)

L = I

Compare to linear momentum:

p = mv

Linear Momentum and ForceLinear Momentum and ForceAngular Momentum and TorqueAngular Momentum and Torque

Linear F =p/to Impulse p = F t

Rotational = L/ to Rotational Impulse L = t

Conservation of MomentumConservation of Momentum

Linearo If F = 0, then p is constant.

Angularo If = 0, then L is constant.

Conservation of Conservation of Angular MomentumAngular Momentum

Ice Skater Throwing a football Rifling Helicopters Precession

RiflingRifling

Football PhysicsFootball Physics

L

Helicopter PhysicsHelicopter Physics

Rotation of Rotor

Body Rotation

Tail rotor used to produce thrust in opposite direction

of body rotation

PrecessionPrecession

Age of AquariusAge of Aquarius

Linear - Rotational ConnectionsLinear - Rotational Connections

Linear Rotational x (m) (rad)

v (m/s) (rad/s)

a (m/s2) (rad/s2)

m (kg) I (kg·m2)F (N) (N·m)

p (N·s) L (N·m·s)