Chapter 9Chapter 9

Rotational MotionRotational Motion

Rotational MotionRotational Motion

Many interesting physical Many interesting physical phenomena are not “linear”phenomena are not “linear”

Ball at the end of a string revolvingBall at the end of a string revolving Planets around SunPlanets around Sun Moon around EarthMoon around Earth

New Physical Quantities:

•Angular Displacement

•Angular velocity

•Angular acceleration

The RadianThe Radian

The radian is a unit The radian is a unit of angular measureof angular measure

The radian can be The radian can be defined as the arc defined as the arc length s along a length s along a circle divided by circle divided by the radius rthe radius r

sr

57.3°

More About RadiansMore About Radians

Whole circleWhole circle Comparing degrees and radiansComparing degrees and radians

Converting from degrees to radiansConverting from degrees to radians

3.572

360rad1

]rees[deg180

]rad[

radrev 23601

ExampleExample

Express it in radians, Express it in radians, revolutions.revolutions.

radrad

855.0deg360

2deg49

revrev

136.0deg360

deg49

revrad

revrad 136.0

2855.0

Angular DisplacementAngular Displacement

Axis of rotation is Axis of rotation is the center of the the center of the diskdisk

Need a fixed Need a fixed reference linereference line

During time t, the During time t, the line on rotating line on rotating object moves object moves through angle θthrough angle θ

Angular Displacement, cont.Angular Displacement, cont.

The The angular displacementangular displacement is defined is defined as the angle the object rotates as the angle the object rotates through during some time intervalthrough during some time interval

The unit of angular displacement is The unit of angular displacement is

the radianthe radian Each point on the object undergoes Each point on the object undergoes

the same angular displacementthe same angular displacement

IF

Average Angular VelocityAverage Angular Velocity

The average The average angular velocity, ω, angular velocity, ω, of a rotating rigid of a rotating rigid object is the ratio object is the ratio of the angular of the angular displacement to displacement to the time intervalthe time interval

ttIF

t

Angular velocity, cont.Angular velocity, cont.

The The instantaneousinstantaneous angular velocity angular velocity Units of angular velocity are Units of angular velocity are

radians/secradians/sec• rad/srad/s

Velocity will be positive if θ is Velocity will be positive if θ is increasing (counterclockwise)increasing (counterclockwise)

Velocity will be negative if θ is Velocity will be negative if θ is decreasing (clockwise)decreasing (clockwise)

Average Angular AccelerationAverage Angular Acceleration

The average angular acceleration The average angular acceleration of an object is defined as the ratio of of an object is defined as the ratio of the change in the angular velocity to the change in the angular velocity to the time it takes for the object to the time it takes for the object to undergo the change:undergo the change:

tIF

tIF

Angular Acceleration, contAngular Acceleration, cont

Units of angular acceleration are rad/s²Units of angular acceleration are rad/s² Positive angular accelerations are in the Positive angular accelerations are in the

counterclockwise direction and negative counterclockwise direction and negative accelerations are in the clockwise directionaccelerations are in the clockwise direction

When a rigid object rotates about a fixed When a rigid object rotates about a fixed axis, every portion of the object has the axis, every portion of the object has the same angular velocity and the same same angular velocity and the same angular accelerationangular acceleration

Angular speedAngular speed

Angular Acceleration, finalAngular Acceleration, final

The sign of the acceleration does not The sign of the acceleration does not have to be the same as the sign of have to be the same as the sign of the angular velocitythe angular velocity

The instantaneous angular The instantaneous angular accelerationacceleration

Only consider rotations with uniform Only consider rotations with uniform angular accelerationangular acceleration

ExampleExample

A bicycle wheel makes 50 revolutions A bicycle wheel makes 50 revolutions in 0.5 min. Compute the angular in 0.5 min. Compute the angular speed in rev/s, rad/s, deg/s.speed in rev/s, rad/s, deg/s.

ExampleExample

The bicycle wheel traveling at 1.67 The bicycle wheel traveling at 1.67 rev/s decelerates uniformly to a stop rev/s decelerates uniformly to a stop in 3 seconds. Compute the angular in 3 seconds. Compute the angular deceleration in rev/s², rad/s², deg/s² deceleration in rev/s², rad/s², deg/s²

Analogies Between Linear and Analogies Between Linear and Rotational MotionRotational Motion

atvv IF

FIaverage vvv 2

1

2

2

1attvx I

x

vva

IF

2

22

tIF

FIaverage 2

1

2

2

1ttI

2

22IF

Linear Motion with constant acc.

(x,v,a)

Rotational Motion with fixed axisand constant

ExampleExample

A car coasts to a stop with a uniform A car coasts to a stop with a uniform deceleration of 1.2 rad/s² for its deceleration of 1.2 rad/s² for its wheel. If its initial angular speed was wheel. If its initial angular speed was

1000 rev/min, how many revolutions 1000 rev/min, how many revolutions does the wheel make before coming does the wheel make before coming to a stop?to a stop?

Relationship Between Angular and Relationship Between Angular and Linear QuantitiesLinear Quantities

DisplacementsDisplacements

SpeedsSpeeds

AccelerationsAccelerations

Every point on the Every point on the rotating object has rotating object has the same angular the same angular motionmotion

Every point on the Every point on the rotating object rotating object does does notnot have the have the same linear motionsame linear motion

rs

rvT

r

tr

t

svT

raT

ExamplesExamples

PulleyPulley Rolling wheelRolling wheel A belt runs on a pulley with diameter A belt runs on a pulley with diameter

5 cm which revolves at a speed of 5 cm which revolves at a speed of 1000 rev/min. What length of belt 1000 rev/min. What length of belt passes over the pulley each second?passes over the pulley each second?

ExampleExample

A 60 cm diameter wheel is rotating @ A 60 cm diameter wheel is rotating @ speed of 3.0 rev/s. How fast is this speed of 3.0 rev/s. How fast is this car going?car going?

ExampleExample

If the masses of Atwood’s machine are If the masses of Atwood’s machine are accelerating at 4.8 m/s² and radius of accelerating at 4.8 m/s² and radius of 0.3 m, what is 0.3 m, what is

Centripetal AccelerationCentripetal Acceleration

An object traveling in a circle, even An object traveling in a circle, even though it moves with a constant though it moves with a constant speed, will have an accelerationspeed, will have an acceleration

The centripetal acceleration is due to The centripetal acceleration is due to the change in the the change in the directiondirection of the of the velocityvelocity

Centripetal Acceleration, cont.Centripetal Acceleration, cont.

Centripetal refers Centripetal refers to “center-seeking”to “center-seeking”

The direction of the The direction of the velocity changesvelocity changes

The acceleration is The acceleration is directed toward directed toward the center of the the center of the circle of motioncircle of motion

Centripetal Acceleration, finalCentripetal Acceleration, final

The magnitude of the centripetal The magnitude of the centripetal acceleration is given byacceleration is given by

• This direction is toward the center of the This direction is toward the center of the circlecircle

r

vac

2

Centripetal Acceleration and Centripetal Acceleration and Angular VelocityAngular Velocity

The angular velocity and the linear The angular velocity and the linear velocity are related (v = rω)velocity are related (v = rω)

The centripetal acceleration can also The centripetal acceleration can also be related to the angular velocitybe related to the angular velocity

must be in rad/smust be in rad/s

2rac

Forces Causing Centripetal Forces Causing Centripetal AccelerationAcceleration

Newton’s Second Law says that the Newton’s Second Law says that the centripetal acceleration is accompanied by centripetal acceleration is accompanied by a forcea force

• F = maF = ma

• FF stands for any force that keeps an object stands for any force that keeps an object following a circular pathfollowing a circular path

Tension in a stringTension in a string GravityGravity Force of frictionForce of friction

r

vmF

2

ExamplesExamples Ball at the end of Ball at the end of

revolving string revolving string (m=500 grams, (m=500 grams, r=120 cm, r=120 cm, rev/s). rev/s). Tension?Tension?

Fast car rounding Fast car rounding a curvea curve

More on circular MotionMore on circular Motion

Length of circumference = 2Length of circumference = 2RR Period T (time for one complete Period T (time for one complete

circle)circle)

2

22 )2(

2

r

r

r

va

v

r

2

24

r

a

Newton’s Law of Universal Newton’s Law of Universal GravitationGravitation

Every particle in the Universe Every particle in the Universe attracts every other particle with a attracts every other particle with a force that is directly proportional to force that is directly proportional to the product of the masses and the product of the masses and inversely proportional to the square inversely proportional to the square of the distance between them.of the distance between them.

221

R

mmGF

Universal Gravitation, 2Universal Gravitation, 2

G is the constant of universal G is the constant of universal gravitationalgravitational

G = 6.673 x 10G = 6.673 x 10-11-11 N m² /kg² N m² /kg² This is an example of an This is an example of an inverse inverse

square lawsquare law

Motion of SatellitesMotion of Satellites

Consider only circular orbitConsider only circular orbit Radius of orbit r:Radius of orbit r: Gravitational force is the centripetal Gravitational force is the centripetal

force.force.

hRr E

22

2 vr

mG

r

vm

r

mmGmaF EE

r

Gmv E

Motion of SatellitesMotion of Satellites

Period Period

Synchronous OrbitsSynchronous Orbits

v

r 2

EGm

r 232 Kepler’s 3rd Law

milesmrm

Gshr

E4724

11

106.21023.4106

,1067.6 ,8640024

Communications SatelliteCommunications Satellite

A geosynchronous orbitA geosynchronous orbit• Remains above the same place on the earthRemains above the same place on the earth• The period of the satellite will be 24 hrThe period of the satellite will be 24 hr

r = h + Rr = h + REE

Still independent of the mass of the satelliteStill independent of the mass of the satellite

milesRrh

mileskmR

milesmr

E

E

000,22

40006370

106.21023.4 47

Satellites and WeightlessnessSatellites and Weightlessness

weighting an object in an elevatorweighting an object in an elevator Elevator at rest: mgElevator at rest: mg Elevator accelerates upward: Elevator accelerates upward:

m(g+a)m(g+a) Elevator accelerates downward: Elevator accelerates downward:

m(g+a) with a<0m(g+a) with a<0 Satellite: a=-g!!Satellite: a=-g!!

Chapter 10Chapter 10

Dynamics of Dynamics of RotationRotation

Force vs. TorqueForce vs. Torque

Forces cause accelerationsForces cause accelerations Torques cause angular accelerationsTorques cause angular accelerations

• rotationrotation Force and torque are relatedForce and torque are related

Torque, contTorque, cont

Torque, Torque, , is the tendency of a force to , is the tendency of a force to rotate an object about some axisrotate an object about some axis

is the torqueis the torque F is the forceF is the force

• symbol is the Greek tausymbol is the Greek tau ll is the lever arm is the lever arm

SI unit is NSI unit is N..mm Lever Arm: Perpendicular distance from Lever Arm: Perpendicular distance from

the pivot point to the line of forcethe pivot point to the line of force

Fl

Direction of TorqueDirection of Torque

Torque is a vector quantityTorque is a vector quantity• We will treat only 2-d torque so no We will treat only 2-d torque so no

need for vector notion.need for vector notion.• If the turning tendency of the force is If the turning tendency of the force is

counterclockwise, the torque will be counterclockwise, the torque will be positive (+)positive (+)

• If the turning tendency is clockwise, If the turning tendency is clockwise, the torque will be negative (-)the torque will be negative (-)

Multiple TorquesMultiple Torques

When two or more torques are acting When two or more torques are acting on an object, the torques are addedon an object, the torques are added• with the signswith the signs

If the net torque is zero, the object’s If the net torque is zero, the object’s rate of rotation doesn’t changerate of rotation doesn’t change

General Definition of TorqueGeneral Definition of Torque

The applied force is not always The applied force is not always perpendicular to the position vectorperpendicular to the position vector

The component of the force The component of the force perpendicularperpendicular to the object will cause to the object will cause it to rotateit to rotate

sin

sin

Ll

FL

F is the forceF is the force

L is distance L is distance between pivot and between pivot and point of actionpoint of action

is the angleis the angle

Moment of InertiaMoment of Inertia

The angular acceleration is inversely The angular acceleration is inversely proportional to the analogy of the proportional to the analogy of the mass in a rotating systemmass in a rotating system

This mass analog is called the This mass analog is called the moment of inertia, moment of inertia, I, of the objectI, of the object

• SI units are kg mSI units are kg m22

2I mr

Newton’s Second Law for a Newton’s Second Law for a Rotating ObjectRotating Object

The angular acceleration is directly The angular acceleration is directly proportional to the net torqueproportional to the net torque

The angular acceleration is inversely The angular acceleration is inversely proportional to the moment of inertia proportional to the moment of inertia of the objectof the object

I

More About Moment of InertiaMore About Moment of Inertia

There is a major difference between There is a major difference between moment of inertia and mass: the moment of inertia and mass: the moment of inertia depends on the moment of inertia depends on the quantity of matter quantity of matter and its distributionand its distribution in the rigid object.in the rigid object.

The moment of inertia also depends The moment of inertia also depends upon the location of the axis of upon the location of the axis of rotationrotation

Moment of Inertia of a Uniform Moment of Inertia of a Uniform RingRing

Image the hoop is Image the hoop is divided into a divided into a number of small number of small segments, msegments, m11 … …

These segments These segments are equidistant are equidistant from the axisfrom the axis

2 2i iI m r MR

Other Moments of InertiaOther Moments of Inertia

ExampleExample

Wheel of radius R=20 cm and Wheel of radius R=20 cm and I=30kg·m^2. Force F=40N acts I=30kg·m^2. Force F=40N acts along the edge of the wheel.along the edge of the wheel.

1.1. Angular acceleration?Angular acceleration?

2.2. Angular speed 4s after starting from Angular speed 4s after starting from rest?rest?

3.3. Number of revolutions for the 4s?Number of revolutions for the 4s?

Rotational Kinetic EnergyRotational Kinetic Energy

An object rotating about some axis An object rotating about some axis with an angular speed, ω, has with an angular speed, ω, has rotational kinetic energy Erotational kinetic energy Ekrkr==½Iω½Iω22

Energy concepts can be useful for Energy concepts can be useful for simplifying the analysis of rotational simplifying the analysis of rotational motionmotion

krktk EEE

Total Energy of a SystemTotal Energy of a System

Conservation of Mechanical EnergyConservation of Mechanical Energy

• Remember, this is for conservative Remember, this is for conservative forces, no dissipative forces such as forces, no dissipative forces such as friction can be presentfriction can be present

• Potential energies of any other Potential energies of any other conservative forces could be addedconservative forces could be added

pFkFpIkI

krktk

EEEE

EEE

Rolling down inclineRolling down incline

Energy conservationEnergy conservation Linear velocity and angular speed are Linear velocity and angular speed are

related v=Rrelated v=R

Smaller I, bigger v, faster!!Smaller I, bigger v, faster!!

22

2

1

2

1 Imvmgh

22

22

2 )(2

1)(

2

1

2

1v

R

Imv

R

Imvmgh

Work-Energy in a Rotating Work-Energy in a Rotating SystemSystem

In the case where there are In the case where there are dissipative forces such as friction, dissipative forces such as friction, use the generalized Work-Energy use the generalized Work-Energy Theorem instead of Conservation of Theorem instead of Conservation of EnergyEnergy

(E(Ektkt+E+Ekrkr+E+Epp))ii++W=(EW=(Ektkt+E+Ektkt+E+Epp))ff

Angular MomentumAngular Momentum

Similarly to the relationship between Similarly to the relationship between force and momentum in a linear force and momentum in a linear system, we can show the relationship system, we can show the relationship between torque and angular between torque and angular momentummomentum

Angular momentum is defined as Angular momentum is defined as • L = I ωL = I ω

• and and Lt

Angular Momentum, contAngular Momentum, cont

If the net torque is zero, the angular If the net torque is zero, the angular momentum remains constantmomentum remains constant

Conservation of Angular MomentumConservation of Angular Momentum states: The angular momentum of a states: The angular momentum of a system is conserved when the net system is conserved when the net external torque acting on the external torque acting on the systems is zero.systems is zero.• That is, when That is, when

0, i fi i ffL L or I I

Conservation Rules, SummaryConservation Rules, Summary

In an isolated system, the following In an isolated system, the following quantities are conserved:quantities are conserved:• Mechanical energyMechanical energy• Linear momentumLinear momentum• Angular momentumAngular momentum

Conservation of Angular Conservation of Angular Momentum, ExampleMomentum, Example

With hands and With hands and feet drawn closer feet drawn closer to the body, the to the body, the skater’s angular skater’s angular speed increasesspeed increases• L is conserved, I L is conserved, I

decreases, decreases, increasesincreases

ExampleExample

A 500 grams uniform sphere of 7.0 cm A 500 grams uniform sphere of 7.0 cm radius spins at 30 rev/s on an axis radius spins at 30 rev/s on an axis through its center.through its center.

Moment of inertiaMoment of inertia Rotational kinetic energyRotational kinetic energy Angular momentumAngular momentum

ExampleExample

A turntable is a uniform disk of metal A turntable is a uniform disk of metal of mass 1.5 kg and radius 13 cm. of mass 1.5 kg and radius 13 cm. What torque is required to drive the What torque is required to drive the turntable so that it accelerates at a turntable so that it accelerates at a constant rate from 0 to 33.3 rpm in 2 constant rate from 0 to 33.3 rpm in 2 seconds?seconds?