Chapter 6,9,10Chapter 6,9,10

Circular Motion, Gravitation, Circular Motion, Gravitation, Rotation, Bodies in Rotation, Bodies in

EquilibriumEquilibrium

Circular MotionCircular Motion

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

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

Comparing degrees and radiansComparing degrees and radians

Converting from degrees to radiansConverting from degrees to radians

3.572

360rad1

]rees[deg180

]rad[

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 reference line reference line moves through moves through angle θangle 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 SpeedAverage Angular Speed

The average The average angular speed, ω, angular speed, 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

Angular Speed, cont.Angular Speed, cont.

The The instantaneousinstantaneous angular speed angular speed Units of angular speed are Units of angular speed are

radians/secradians/sec• rad/srad/s

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

Speed will be negative if θ is Speed 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 speed to the change in the angular speed to the time it takes for the object to the time it takes for the object to undergo the change:undergo the change:

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 speed and the same angular same angular speed and the same angular accelerationacceleration

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 speedthe angular speed

The instantaneous angular The instantaneous angular accelerationacceleration

Analogies Between Linear and Analogies Between Linear and Rotational MotionRotational Motion

atvv 0

vvvaverage 02

1

200 2

1attvxx

)(2 0

20

2

xx

vva

t 0

02

1average

200 2

1tt

)(2 0

20

2

Linear Motion with constant acc.

(x,v,a)

Rotational Motion with fixed axisand constant

ExamplesExamples

78 rev/min=?78 rev/min=? A fan turns at a rate of 900 rpmA fan turns at a rate of 900 rpm Tangential speed of tips of 20cm long Tangential speed of tips of 20cm long

blades?blades?

Now the fan is uniformly accelerated Now the fan is uniformly accelerated to 1200 rpm in 20 sto 1200 rpm in 20 s

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

Rx Rv

Ra //

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

va

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 = ωR)velocity are related (v = ωR)

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

Ra 2

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 Ball at the

end of end of revolving revolving stringstring

Fast car Fast car rounding a rounding a curvecurve

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

ExampleExample

200 grams mass revolving in uniform 200 grams mass revolving in uniform circular motion on an horizontal circular motion on an horizontal frictionless surface at 2 revolutions/s. frictionless surface at 2 revolutions/s. What is the force on the mass by the What is the force on the mass by the string (R=20cm)?string (R=20cm)?

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

Universal Gravitation, 3Universal Gravitation, 3

The force that The force that mass 1 exerts on mass 1 exerts on mass 2 is equal mass 2 is equal and opposite to the and opposite to the force mass 2 force mass 2 exerts on mass 1exerts on mass 1

The forces form a The forces form a Newton’s third law Newton’s third law action-reactionaction-reaction

Universal Gravitation, 4Universal Gravitation, 4

The gravitational force exerted by a The gravitational force exerted by a uniform sphere on a particle outside uniform sphere on a particle outside the sphere is the same as the force the sphere is the same as the force exerted if the entire mass of the exerted if the entire mass of the sphere were concentrated on its sphere were concentrated on its centercenter

Gravitation ConstantGravitation Constant

Determined Determined experimentallyexperimentally

Henry CavendishHenry Cavendish• 17981798

The light beam and The light beam and mirror serve to mirror serve to amplify the motionamplify the motion

Applications of Universal Applications of Universal GravitationGravitation

Weighing the EarthWeighing the Earth

G

gRm

R

mGg

R

mmGmg

R

mmGFw

EE

E

E

E

E

E

Eg

2

2

2

2

kg 106

6380

/ 8.9 take

24

2

E

E

m

kmR

smg

Applications of Universal Applications of Universal GravitationGravitation

““g” will vary with g” will vary with altitudealtitude

2""

r

mGg E

Escape SpeedEscape Speed

The escape speed is the speed The escape speed is the speed needed for an object to soar off into needed for an object to soar off into space and not returnspace and not return

For the earth, vFor the earth, vesc esc is about 11.2 km/sis about 11.2 km/s Note, v is independent of the mass of Note, v is independent of the mass of

the objectthe object

E

Eesc R

Gmv

2

Various Escape SpeedsVarious Escape Speeds

The escape speeds The escape speeds for various for various members of the members of the solar systemsolar system

Escape speed is Escape speed is one factor that one factor that determines a determines a planet’s planet’s atmosphereatmosphere

Motion of SatellitesMotion of Satellites

Consider only Consider only circular orbitcircular orbit

Radius of orbit r:Radius of orbit r: Gravitational force Gravitational force

is the centripetal is the centripetal force.force.

hRr E

22

2 vr

mG

r

vm

r

mmGmaF

EE

r

Gmv

E

r

Motion of SatellitesMotion of Satellites

Period Period

v

r 2

EGm

r 232 Kepler’s 3rd Law

milesmrm

Gs

E4724

11

106.21023.4106

,1067.6 ,86400

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

milesmrm

Gs

E4724

11

106.21023.4106

,1067.6 ,86400

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!!

Force vs. TorqueForce vs. Torque

Forces cause accelerationsForces cause accelerations Torques cause angular accelerationsTorques cause angular accelerations Force and torque are relatedForce and torque are related

TorqueTorque

The door is free to rotate about an axis through OThe door is free to rotate about an axis through O There are three factors that determine the There are three factors that determine the

effectiveness of the force in opening the door:effectiveness of the force in opening the door:• The The magnitudemagnitude of the force of the force• The The positionposition of the application of the force of the application of the force• The The angleangle at which the force is applied at which the force is applied

Torque, contTorque, cont

Torque, Torque, , is the tendency of a force , is the tendency of a force to rotate an object about some axisto rotate 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 l is the length of lever arml is the length of lever arm

SI unit is NSI unit is N..mm Work done by torque W=Work done by torque W=

Fl

Direction of TorqueDirection of Torque

If the turning tendency of the force If the turning tendency of the force is counterclockwise, the torque will is counterclockwise, the torque will be positivebe positive

If the turning tendency is If the turning tendency is clockwise, the torque will be clockwise, the torque will be negativenegative

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

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

Torque and EquilibriumTorque and Equilibrium

First Condition of EquilibriumFirst Condition of Equilibrium The net external force must be zeroThe net external force must be zero

• This is a necessary, but not sufficient, condition This is a necessary, but not sufficient, condition to ensure that an object is in complete to ensure that an object is in complete mechanical equilibriummechanical equilibrium

• This is a statement of translational equilibriumThis is a statement of translational equilibrium

0

0 0x y

or

and

F

F F

Torque and Equilibrium, contTorque and Equilibrium, cont

To ensure mechanical equilibrium, To ensure mechanical equilibrium, you need to ensure rotational you need to ensure rotational equilibrium as well as translationalequilibrium as well as translational

The Second Condition of Equilibrium The Second Condition of Equilibrium statesstates• The net external torque must be zeroThe net external torque must be zero

0

Equilibrium ExampleEquilibrium Example

The woman, mass m, The woman, mass m, sits on the left end of sits on the left end of the see-sawthe see-saw

The man, mass M, sits The man, mass M, sits where the see-saw will where the see-saw will be balancedbe balanced

Apply the Second Apply the Second Condition of Condition of Equilibrium and solve Equilibrium and solve for the unknown for the unknown distance, xdistance, x

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·m². Force F=40N acts along I=30kg·m². Force F=40N acts along the edge of the wheel.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?

4.4. Work done on the wheel?Work done on the wheel?

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 KErotational kinetic energy KErr==½Iω½Iω22

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

Units (rad/s)!!Units (rad/s)!!

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

( ) ( )t r g i t r g fKE KE PE KE KE PE

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

(KE(KEtt+KE+KERR+PE)+PE)ii++W=(KEW=(KEtt+KE+KERR+PE)+PE)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

Find work done to open 30Find work done to open 30 a 1m wide a 1m wide door with a steady force of 0.9N at door with a steady force of 0.9N at right angle to the surface of the door.right angle to the surface of the door.

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?

Center of GravityCenter of Gravity

The force of gravity acting on an The force of gravity acting on an object must be consideredobject must be considered

In finding the torque produced by the In finding the torque produced by the force of gravity, all of the weight of force of gravity, all of the weight of the object can be considered to be the object can be considered to be concentrated at a single pointconcentrated at a single point

Calculating the Center of Calculating the Center of GravityGravity

The object is The object is divided up into a divided up into a large number of large number of very small particles very small particles of weight (mg)of weight (mg)

Each particle will Each particle will have a set of have a set of coordinates coordinates indicating its indicating its location (x,y)location (x,y)

Calculating the Center of Calculating the Center of Gravity, cont.Gravity, cont.

We wish to locate the point of We wish to locate the point of application of the application of the single forcesingle force whose whose magnitude is equal to the weight of magnitude is equal to the weight of the object, and whose effect on the the object, and whose effect on the rotation is the same as all the rotation is the same as all the individual particles.individual particles.

This point is called the This point is called the center of center of gravitygravity of the object of the object

Coordinates of the Center of Coordinates of the Center of GravityGravity

The coordinates of the center of The coordinates of the center of gravity can be foundgravity can be found

i i i icg cg

i i

m x m yx and y

m m

Center of Gravity of a Uniform Center of Gravity of a Uniform ObjectObject

The center of gravity of a The center of gravity of a homogenous, symmetric body must homogenous, symmetric body must lie on the axis of symmetry.lie on the axis of symmetry.

Often, the center of gravity of such Often, the center of gravity of such an object is the an object is the geometricgeometric center of center of the object.the object.

ExampleExample

Find the center of mass (gravity) of Find the center of mass (gravity) of these masses: 3kg (0,1), 2kg (0,0)these masses: 3kg (0,1), 2kg (0,0)

And 1kg (2,0)And 1kg (2,0)

ExampleExample

Find the center of mass (gravity) of the Find the center of mass (gravity) of the dumbbell, 4 kg and 2 kg with a 4m dumbbell, 4 kg and 2 kg with a 4m long 3kg rod.long 3kg rod.

Torque, reviewTorque, review

is the torqueis the torque FF is the force is the force

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

SI unit is NSI unit is N..mm

Fl

Direction of TorqueDirection of Torque

If the turning tendency of the force If the turning tendency of the force is counterclockwise, the torque will is counterclockwise, the torque will be positivebe positive

If the turning tendency is If the turning tendency is clockwise, the torque will be clockwise, the torque will be negativenegative

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

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

ExampleExample

A 2 m by 2 m square metal plate A 2 m by 2 m square metal plate rotates about its center. Calculate rotates about its center. Calculate the torque of all five forces each with the torque of all five forces each with magnitude 50N.magnitude 50N.

Torque and EquilibriumTorque and Equilibrium

First Condition of EquilibriumFirst Condition of Equilibrium The net external force must be zeroThe net external force must be zero

0

0 0x y

or

and

F

F F

The Second Condition of Equilibrium The Second Condition of Equilibrium statesstates

•The net external torque must be zeroThe net external torque must be zero

0

ExampleExample

The system is in equilibrium. Calculate The system is in equilibrium. Calculate W and find the tension in the rope W and find the tension in the rope (T).(T).

ExampleExample

A 160 N boy stands on a 600 N A 160 N boy stands on a 600 N concrete beam in equilibrium with concrete beam in equilibrium with two end supports. If he stands one two end supports. If he stands one quarter the length from one support, quarter the length from one support, what are the forces exerted on the what are the forces exerted on the beam by the two supports?beam by the two supports?