Chapters 7 & 8Chapters 7 & 8

Rotational MotionRotational Motion

andand

The Law of GravityThe Law of Gravity

HomeworkHomework

Problems:Problems: CH 7: CH 7:

1,4,5,7,10,12,14,18,19,21,24,25,29,31,4,5,7,10,12,14,18,19,21,24,25,29,31,34,36,401,34,36,40

CH8: CH8: 17,21, 2317,21, 23

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, Angular Displacement, cont.cont.

Every point on the object undergoes Every point on the object undergoes circular motion about the point Ocircular motion about the point O

Angles generally need to be measured Angles generally need to be measured in in radiansradians

s is the length of arc and r is the s is the length of arc and r is the radiusradius

r

v

tr

s

t

r

s

rvt

More About RadiansMore About Radians

Comparing degrees and radiansComparing degrees and radians

Converting from degrees to Converting from degrees to radiansradians

3.572

360rad1

]rees[deg180

]rad[

Angular Displacement, Angular Displacement, cont.cont.

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

Every point on the disc undergoes Every point on the disc undergoes the same angular displacement in the same angular displacement in any given time intervalany given time interval

Angular SpeedAngular Speed

The The average average angular speed, ωangular 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

ttt if

if

r

v

t

vac

2

Angular Speed, cont.Angular Speed, cont.

The The instantaneousinstantaneous angular speed is angular speed is defined as the limit of the average defined as the limit of the average speed as the time interval approaches speed as the time interval approaches zerozero

Units of angular speed are radians/secUnits of angular speed are radians/sec rad/srad/s

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

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

Angular AccelerationAngular Acceleration

The average The average angular acceleration, angular acceleration, , ,

of an object is defined as the ratio of an object is defined as the ratio of the change in the angular speed of 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:to undergo the change:

ttt if

if

Tangential AccelerationTangential Acceleration

rvt rvt

tr

t

vt

rat

More About Angular More About Angular AccelerationAcceleration

Units of angular acceleration are Units of angular acceleration are rad/s²rad/s²

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

Problem Solving HintsProblem Solving Hints

Similar to the techniques used in linear Similar to the techniques used in linear motion problemsmotion problems With constant angular acceleration, the With constant angular acceleration, the

techniques are much like those with constant techniques are much like those with constant linear accelerationlinear acceleration

There are some differences to keep in There are some differences to keep in mindmind For rotational motion, define a rotational axisFor rotational motion, define a rotational axis The object keeps returning to its original The object keeps returning to its original

orientation, so you can find the number of orientation, so you can find the number of revolutions made by the bodyrevolutions made by the body

Analogies Between Linear Analogies Between Linear and Rotational Motionand Rotational Motion

Rotational Motion Rotational Motion About a Fixed Axis About a Fixed Axis with Constant with Constant AccelerationAcceleration

Linear Motion with Linear Motion with Constant Constant AccelerationAcceleration

ti 2

i t2

1t

22i

2xa2vv 2

i2

2i at

2

1tvx

atvv i

Relationship Between Relationship Between Angular and Linear Angular and Linear QuantitiesQuantities

DisplacementsDisplacements

SpeedsSpeeds

AccelerationsAccelerations

Every point on the Every point on the rotating object rotating object has the same has the same angular motionangular motion

Every point on the Every point on the rotating object rotating object does does notnot have the have the same linear same linear motionmotion

rs

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 The centripetal acceleration is due to the change in the to the change in the directiondirection of of the velocitythe velocity

Centripetal Acceleration, Centripetal Acceleration, cont.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 the directed toward the center of the circle center of the circle of motionof motion

Centripetal Acceleration Centripetal Acceleration and Angular Velocityand Angular 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 2C

r

vac

2

See page 198 for derivation

OR

Total AccelerationTotal Acceleration The tangential component of the The tangential component of the

acceleration is due to changing speedacceleration is due to changing speed The centripetal component of the The centripetal component of the

acceleration is due to changing directionacceleration is due to changing direction Total acceleration can be found from Total acceleration can be found from

these componentsthese components

2C

2t aaa

Pythagorean theorem

Vector Nature of Angular Vector Nature of Angular QuantitiesQuantities

Assign a positive or Assign a positive or negative direction in negative direction in the problemthe problem

A more complete way A more complete way is by using the right is by using the right hand rulehand rule Grasp the axis of Grasp the axis of

rotation with your right rotation with your right handhand

Wrap your fingers in Wrap your fingers in the direction of rotationthe direction of rotation

Your thumb points in Your thumb points in the direction of ωthe direction of ω

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 centripetal acceleration is accompanied by a forceby a forceF stands for any force that keeps an object F 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

2

22 4

T

rm

r

mvmaFF ccnet

Problem Solving StrategyProblem Solving Strategy

Draw a free body diagram, Draw a free body diagram, showing and labeling all the forces showing and labeling all the forces acting on the object(s)acting on the object(s)

Choose a coordinate system that Choose a coordinate system that has one axis perpendicular to the has one axis perpendicular to the circular path and the other axis circular path and the other axis tangent to the circular pathtangent to the circular path

Problem Solving Strategy, Problem Solving Strategy, cont.cont.

Find the net force toward the center of Find the net force toward the center of the circular path (this is the force that the circular path (this is the force that causes the centripetal acceleration)causes the centripetal acceleration)

Solve as in Newton’s second law Solve as in Newton’s second law problemsproblems The directions will be radial and tangentialThe directions will be radial and tangential The acceleration will be the centripetal The acceleration will be the centripetal

accelerationacceleration

Applications of Forces Applications of Forces Causing Centripetal Causing Centripetal AccelerationAcceleration

Many specific situations will use Many specific situations will use forces that cause centripetal forces that cause centripetal accelerationacceleration Level curvesLevel curves Banked curvesBanked curves Horizontal circlesHorizontal circles Vertical circlesVertical circles

Level CurvesLevel Curves

Friction is the Friction is the force that force that produces the produces the centripetal centripetal accelerationacceleration

Can find the Can find the frictional force, µ, frictional force, µ, vv rgv

mgf

nfr

vmf

maf

2

But what about the mass???

r

vmmg

2

At what At what maximum maximum speed can a car speed can a car negotiate a negotiate a turn on a wet turn on a wet road with road with coefficient of coefficient of static friction static friction 0.230 without 0.230 without sliding out of sliding out of control?control?

Banked CurvesBanked Curves

A component of A component of the normal force the normal force adds to the adds to the frictional force to frictional force to allow higher allow higher speedsspeeds

rg

v2

tan

tanremember

cos

sinfrom

See p. 204

A race track is to have a banked A race track is to have a banked curve with a radius of 25m. What curve with a radius of 25m. What should be the angle of the bank if the should be the angle of the bank if the normal force alone is to allow safe normal force alone is to allow safe travel around the curve at 58.0 m/s?travel around the curve at 58.0 m/s?

Horizontal CircleHorizontal Circle

The horizontal The horizontal component of the component of the tension causes tension causes the centripetal the centripetal accelerationacceleration

tangaC

See next page for derivation

DerivationDerivation

tantan

tancos

sinsin

cos

0cos

gm

gm

m

Fa

maF

mgmg

TF

mgT

mgT

c

c

What

about

mass???

Vertical CircleVertical Circle

Look at the forces Look at the forces at the top of the at the top of the circlecircle

The minimum The minimum speed at the top speed at the top of the circle can of the circle can be foundbe foundgRv top

See ex. 7.9 on page 205

A jet traveling at a speed of 1.20 x 10A jet traveling at a speed of 1.20 x 102 2 m/s executes a vertical loop with a m/s executes a vertical loop with a radius 5.00 x 10radius 5.00 x 102 2 m. Find the m. Find the magnitude of the force of the seat on a magnitude of the force of the seat on a 70.0 kg pilot (a) at the top (b)the 70.0 kg pilot (a) at the top (b)the bottom of the loop.bottom of the loop.

Forces in Accelerating Forces in Accelerating Reference FramesReference Frames

Distinguish real forces from Distinguish real forces from fictitious forcesfictitious forces

Centrifugal force is a fictitious Centrifugal force is a fictitious forceforce

Real forces always represent Real forces always represent interactions between objectsinteractions between objects

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

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

221

r

mmGF

221

G r

mmGF or

Law of GravitationLaw of Gravitation

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

G = 6.673 x 10G = 6.673 x 10-11-11 N m² /kg²N m² /kg²

This is an This is an example of an example of an inverse square inverse square lawlaw

Gravitation ConstantGravitation Constant

Determined Determined experimentallyexperimentally

Henry CavendishHenry Cavendish 17981798

The light beam The light beam and mirror serve and mirror serve to amplify the to amplify the motionmotion

Applications of Universal Applications of Universal GravitationGravitation

Mass of the earthMass of the earth Use an example Use an example

of an object close of an object close to the surface of to the surface of the earththe earth r ~ Rr ~ REE

G

gRM

2E

E

Applications of Universal Applications of Universal GravitationGravitation

Acceleration due Acceleration due to gravityto gravity

g will vary with g will vary with altitudealtitude

2E

r

MGg

Gravitational Gravitational Field LinesField Lines

Gravitational Field Gravitational Field Strength is Strength is considered force considered force per unit massper unit mass

2

2

2

r

GM

m

Fg

r

GM

m

Fr

mGmF

earth

earth

testmass

testmassearth

Gravitational Potential Gravitational Potential EnergyEnergy

PE = mgy is PE = mgy is valid only near valid only near the earth’s the earth’s surfacesurface

For objects high For objects high above the above the earth’s surface, earth’s surface, an alternate an alternate expression is expression is neededneeded

Zero reference Zero reference level is infinitely level is infinitely far from the far from the earthearth

r

mMGPE E

r

GmmPE

r

Gmmmgr

PEmgrr

Gmmmg

mghPE

mgFr

mGmF

2

2

22

2

21

Einstein’s view of Gravity Einstein’s view of Gravity Space-TimeSpace-Time

Kepler’s LawsKepler’s Laws

All planets move in elliptical orbits All planets move in elliptical orbits with the Sun at one of the focal with the Sun at one of the focal points.points.

A line drawn from the Sun to any A line drawn from the Sun to any planet sweeps out equal areas in planet sweeps out equal areas in equal time intervals.equal time intervals.

The square of the orbital period of The square of the orbital period of any planet is proportional to cube of any planet is proportional to cube of the average distance from the Sun the average distance from the Sun to the planet.to the planet.

Kepler’s Laws, cont.Kepler’s Laws, cont.

Based on observations made by Based on observations made by BraheBrahe

Newton later demonstrated that Newton later demonstrated that these laws were consequences of these laws were consequences of the gravitational force between the gravitational force between any two objects together with any two objects together with Newton’s laws of motionNewton’s laws of motion

Kepler’s First LawKepler’s First Law

All planets move All planets move in elliptical orbits in elliptical orbits with the Sun at with the Sun at one focus.one focus. Any object bound Any object bound

to another by an to another by an inverse square law inverse square law will move in an will move in an elliptical pathelliptical path

Second focus is Second focus is emptyempty

Kepler’s Second LawKepler’s Second Law

A line drawn from A line drawn from the Sun to any the Sun to any planet will sweep planet will sweep out equal areas in out equal areas in equal timesequal times Area from A to B Area from A to B

and C to D are the and C to D are the samesame

Kepler’s Third LawKepler’s Third Law

The square of the orbital The square of the orbital period of any planet is period of any planet is proportional to cube of proportional to cube of the average distance the average distance from the Sun to the from the Sun to the planet.planet.

For orbit around the Sun, For orbit around the Sun, KKSS = 2.97x10 = 2.97x10-19-19 s s22/m/m33

K is independent of the K is independent of the mass of the planetmass of the planet

32 KrT 32

b

a

b

a

r

r

T

T

DerivationDerivation

332

2

2

2

2

2

4

2

rKrGM

T

rTr

r

MG

r

vMma

r

MMGF

ss

s

pcps

3

2192

1097.24

msx

GMK

ss

So…

Kepler’s Third Law Kepler’s Third Law applicationapplication

Mass of the Sun Mass of the Sun or other celestial or other celestial body that has body that has something something orbiting itorbiting it

Assuming a Assuming a circular orbit is a circular orbit is a good good approximationapproximation