Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley

PowerPoint® Lectures forUniversity Physics, Twelfth Edition

– Hugh D. Young and Roger A. Freedman

Lectures by James Pazun

Midterm 3 Review (Ch 9-14)

Ch 9 Overview: Rotational Motion

• Take our results from “linear” physics and do the same for “angular” physics

• Analogue of – Position ←– Velocity ←– Acceleration ←– Force ←– Mass ←– Momentum ←– Energy ←

Cha

pter

s 1-3

Cha

pter

s 4-7

Ch 9: Rotational Kinematics

l R atan vtan

t R arad

v2tan

R R 2vtan

lt

R

Relation between linear quantities and angular quantities (by geometry)

Kinematic equation just like before:θ like displacement xω like velocity vα like acceleration a

Kinetic energy of a rotating object about a fixed (stationary) axis:

KErot 12

I 2 I miri2

i Moment of inertia or “angular mass”

(DEPENDS ON THE AXIS CHOSEN)

KE 12

mvcm2

12

Icm2 Sum of translational and rotational energies

Rolling without Slipping• In reality, car tires both rotate and

translate• They are a good example of

something which rolls (translates, moves forward, rotates) without slipping

• Is there friction? What kind?SM Flag: 30%Relation between translational and rotational velocities in no slip condition

A Rolling Wheel

• A wheel rolls on the surface without slipping with velocity V (your speedometer)

• What is the velocity of the center of the wheel (point C)?

• What is the velocity of the lowest point (point P) w.r.t. the ground?– Does it make sense to you?

Try Differently: Paper Roll• A paper towel unrolls

with velocity V– Conceptually same thing

as the wheel– What’s the velocity of

points: – A? B? C? D? ABC

C B A

D• Point C is where rolling

part separates from the unrolled portion– Both have same velocity

there

Bicycle comes to RestA bicycle with initial linear velocity V0 decelerates

uniformly (without slipping) to rest over a distance d. For a wheel of radius R:

a) What is the angular velocity at t0=0?b) Total revolutions before it stops?c) Total angular distance traversed

by wheel?d) The angular acceleration?e) The total time until it stops?

An athlete throwing the discusA discus thrower moves the discus in a circle of radius 80.0 cm. At a certain instant the thrower is spinning at an angular speed of 10.0 rad/s and the angular speed is increasing at 50.0 rad/s2. At this instant find the acceleration of the discus and its magnitude

atan r (0.800 m)(50.0 rad/s2) = 40.0 m/s2

arad r 2 (10.0 rad/s)2(0.800 m) = 80.0 m/s2

a atan2 arad

2 89.4 m/s2

Finding the moment of inertia for common shapes: Moment of Inertia will be provided for the exam

What is the velocity of the block when it hits the ground?

The work done by the cable is zero since the two tension forces cancel each other out so energy is conserved

KEi PEi KE f PE f

0 mgh 12

mv2 12

I 2 0

0 mgh 12

mv2 12

12

MR2

vR

2

0

v =2mgh

(m +M/2)

Rolling without slipping

KE 12

mvcm2

12

Icm2

12

mvcm2

12

Icm

R2 v2

When rolling without slipping then vcm R This is the condition to roll without slipping.

Then, if you are rolling without slipping the kinetic energy is

Also note that when one is rolling without slipping (i.e. rolling down an incline) the friction force is static so no work is done by it and energy is conserved in this case.

Consider the speed of a yo-yo toyWhat is the speed of the Yo-yo at the bottom (use conservation of energy)

Why conservation of energy: the hand is not moving so it does no work on the system. You may be confused about the tension but keep in mind that it is an internal force so the sum of the upper and lower tension is zero.

Ei E f

0 Mgh 12

Mvcm2

12

Icm2

0 Mgh 12

Mvcm2

12

12

MR2

vcm

R

2

Mgh 34

Mvcm2

vcm 43

gh

Ch 10: Rotational Dynamics

The direction of the torque is given by the RHR

Torque results from force applied at a distance from a pivot point

sinrFr r r

r F

Torque• Write Torque as

• To find the direction of the torque, wrap your fingers in the direction the torque makes the object twist

• If the axis is fixed, what is the net Torque on the wheel?

Fr

Fr

sin||||||

Torque and Moment of Inertia• Force vs. Torque

F=ma and = I

• Mass vs. Moment of Inertia

dmrI

mrIm2

2

or

Flywheel problem from Ch 9(using work energy theorem)

The cable is wrapped around a cylinder. If it unwinds 2.0 m by pulling it with a force of 9.0 N and it starts at rest, what is its final angular velocity and velocity of the cable? (use work energy theorem)

Wtotal KE f KEi

Fx 12

I 2 0

2Fx

I

4FxmR2 20 rad/s

v R (20 rad/s)(0.060 m) =1.2 m/s

Flywheel problem using torque(using work energy theorem)

The cable is wrapped around a cylinder. If it unwinds 2.0 m by pulling it with a force of 9.0 N and it starts at rest, what is its final angular velocity and velocity of the cable? (use work energy theorem)

FR (9.0 N)(0.06 m)

I 12

MR2

I I

2FRMR2

2FMR

6.0 rad/s2

Use α to get acceleration of the cable:

atan R (0.06m)(6.0 rad/s2) 0.36 m/s2

v2 v02 2atan (x x0)Then use kinematics

v 0 2(0.36 m/s2)(2 m) 1.2 m/s

What is the velocity of the block when it hits the ground?

The work done by the cable is zero since the two tension forces cancel each other out so energy is conserved

KEi PEi KE f PE f

0 mgh 12

mv2 12

I 2 0

0 mgh 12

mv2 12

12

MR2

vR

2

0

v =2mgh

(m +M/2)

Another look at the unwinding cableWhat is the linear acceleration of the block?

These are two coupled objects; one rotates and the other moves linearly

For the rotating wheel we have:

I

TR 12

MR2 aR

T

12

Ma

For the block we have:

mg T ma

mg 12

Ma ma

a mg

m M /2

Combine the two equations to get

SM Flag: 55%Connection between rotational and translational accln of coupled objects

SM Flag: 50%Angular momentum both rxp and Iw

Conservation of angular momentum

Before we saw that if the external forces on a system are zero then linear momentum is conserved. Similar for angular momentum.

If the external torques on a system are zero then the TOTAL angular momentum of the systems is conserved.

Note that in this case the TORQUE has to be zero. There can be still forces acting on the system but they do not generate any torque (e.g. force due to gravity on the cat which acts at the center of mass)

I11 I2 2

r r r

r F Recall that torque was defined as

Similarly the angular momentum of a particle is

r L r r r p r r m r v

How a car’s clutch work

The clutch disk and the gear disk is pushed into each other by two forces that do not impart any torque, what is the final angular velocity when they come together?

Lzbefore Lzafter

IAA IBB (IA IB ) final

final IAA IBB

(IA IB )

Ch 11: Conditions for equilibrium

• Net Force is zero (x,y,z)• Net torque is zero.

Strategy

Fx 0

Fy 0 0

1st Draw the free body diagram and the location where the forces are applied

2nd Choose a Pivot Point which will delete the largest amount of torques

3rd Compute the torques from each force

4th Solve the equations

Problem 11.27The horizontal beam in the figure weighs 150 N, and its center of gravity is at its center. Find the tension in the cable and on the beam at the hinge.

SM Flag: 30%Contact forces

Ch 13: Newton’s Law of Gravitation• The gravitational force

is always attractive and depends on both the masses of the bodies involved and their separations.

Fg Gm1m2

r2

G 6.6742 10-11 N m2

kg2

Weight (skip Weight Watchers, just climb upward)

• Gravity (and hence, weight) decreases as altitude rises.

Gravitational potential energy• Objects changing their distance from earth are also

changing their potential energy with respect to earth.

PE Gm1m2

r12

This is the true potential energy. The zero level is set when they are very far apart

Satellite circular motion

The force is radial so Newton’s 2nd law reads:

r : G ME ms

R2 msv2

R ms

4 2RT 2

v GME

R

T 4 2R3

GME

T R3 / 2

SM Flag: 40%Gravitational force provides centripetal accln

Escaping from the EarthWhat is the velocity you need to shoot straight up from the surface of the earth an not come back (conservation of energy)?

KE1 PE1 KE PE

12

m v2GmME

RE

0 0

v 2GME

RE

1.12 104 m/s = 25,000 mph

SM Flag: 40%Energy conservation

Ch 14: Simple Harmonic Motion• The spring drives the glider back

and forth on the air-track and you can observe the changes in the free-body diagram as the motion proceeds from –A to A and back.

The pendulum is another example

Elements of harmonic motion:

Frequency: f=1/TAngular frequency: ω=2πfAmplitude: maximum deflection/stretch/compression

Simple harmonic motion• An ideal spring

responds to stretch and compression linearly, obeying Hooke’s Law.

Fx kxThe restoring force towards the equilibrium point is LINEAR from the displacement from equilibrium

Let’s look at Newton’s 2nd law:

Fx max kx max md2xdt 2

d2xdt 2

km

x

d2xdt 2 2x

Watch variables change for a glider example • As the glider undergoes

SHM, you can track changes in velocity and acceleration as the position changes between the turning points.

x(t) Acos(t )v(t) Asin(t )a(t) A 2 cos(t )

Energy in SHM• Energy is conserved during SHM and the forms

(potential and kinetic) interconvert as the position of the object in motion changes.

E 12

mvx2

12

kx 2 12

kA2 12

mvmax2

SM Flag: 37%Energy at different points

Energy in SHM II

The simple pendulum

Ftan mgsin mg

Ok when Θ<<1

Ftan matan mLd2dt 2

mLd2dt 2 mg

d2dt 2

gL

gL

f 1

2gL

T 2Lg

The physical pendulum• A physical pendulum

is any real pendulum that uses an extended body in motion. This illustrates a physical pendulum.

T 2I

mgd