Friday, October 16, 1998

Chapter 7: periods and frequencies Centripetal acceleration & force Newton’s Law of Gravity

Hint: Be able to do the homework (both theproblems to turn in AND the recommended ones)you’ll do fine on the exam!

Friday, October 23, 1998 in classChapters 5 - 7 inclusive

You may bring one 3”X5” index card (hand-writtenon both sides), a pencil or pen, and a scientificcalculator with you.

I will put any constants, math, and Ch. 1 - 4formulas which you might need on a singlepage attached to the back of the exam.

t

Angular Velocity

[ ][ ]

[ ]

t

[ ] rad

ss 1

t

Angular Acceleration

[ ][ ]

[ ]

t

[ ] rad / s

ss 2

For an object moving in a circle with a constantangular velocity (), we can define a frequencyand period associated with the motion.

The period of therotation is simplythe time requiredfor the object to goaround the circleexactly one time.

T 2

The frequency is thenumber of revolutionscompleted each second.

f 2

Notice that frequencyand period are inverselyrelated to one another…That is...

fT

1

Period

frequency

T 2

[ ][ ]

[ ]T

2

[ ]/

T rad

rad ss

f 2

[ ][ ]

[ ]f

2

[ ]/

f rad s

rads 1

Equations for Systems InvolvingRotational Motion with Constant

Angular Acceleration

Again, these are completely analogous to whatwe derived for the kinetic equations of a linearsystem with constant linear acceleration!

0 01

22t t

0 t

A wheel rotates with constant angularacceleration of 0 = 2 rad/s2. If thewheel starts from rest, how manyrevolutions does it make in 10 s?

0 01

22t t

01

22 1

2 220 2 10t t ( )( )rad

s s

100 rad 1 2 rev rad

( ) .100 15 91

2

50 rad rev rev rev

rad

s r r f i ( )

We’ve seen how arc lengthrelates to an angle swept out:

Let’s look at how our angular velocityand acceleration relate to linear quantities.

First, a question...

In which direction does the instantaneousvelocity of an object moving in a circle point?

For an object movingin a circle with a constantlinear speed (a constantangular velocity), theinstantaneous velocityvectors are always tangentto the circle of motion!

The magnitude of the tangential velocity can befound from our relationship of arc length to angle...

s r s

trt

v rtan

For an object movingin a circle with a constantlinear speed (a constantangular velocity), theinstantaneous tangentialacceleration is ZERO!

The tangential acceleration tells us how thetangential velocity changes. It will pointeither parallel or anti-parallel to the tangentialvelocity vector.

Under what circumstances will the tangentialacceleration be NON-ZERO?

For an object movingaround a circle with achanging angular velocity,and hence a changingtangential velocity, theinstantaneous tangentialacceleration is NON-ZERO!

Let’s look at how the tangential velocity changeswith time in such a case:

v rtan v

t

v v

t t

r r

t tf i

f i

f i

f i

tan tan tan

v

t

v v

t t

r r

t tf i

f i

f i

f i

tan tan tan

ar

trtan

A wheel of radius 0.1 m rotates withconstant angular acceleration of0 = 2 rad/s2. If the wheel starts fromrest, what is the tangential accelerationat t = 10 s?

a rtan ( . )( ) . 01 2 0 22 2 m rad / s m / s

The tangential acceleration, however, is not the only acceleration we need to consider in problems of circular motion...

The blue arrows represent the direction ofthe CENTRIPETAL ACCELERATION,which always points towards the centerof the circle.

Although our objects movesin a circle at constant speed,it still accelerates.

WHY?

Recall that acceleration and velocityare both vector quantities. Sinceacceleration is a change in velocityover some change in time, an objectwhose velocity changes direction isaccelerating!

Recall our definition ofacceleration:

av v

t t

v

tf i

f i

You can demonstrateto yourself (and thebook provides a niceproof on p. 186)

av

r

r

rrc

t 2 2

2( )

So, tangential acceleration occurs when theangular velocity of an object changes (i.e.,the angular acceleration is not equal to 0).

Centripetal acceleration exists for any objectmoving in a circle. The instantaneouscentripetal acceleration is related to theinstantaneous tangential velocity asdefined before. Centripetal accelerationalways points towards the center of thecircle.

The total acceleration willbe the vector sum of thetangential acceleration andthe centripetal acceleration.

atan

ac

atot

Centripetalacceleration

vector

Tangentialacceleration

a a atot c tan2 2

When an object experiences a centripetalacceleration, there must be a force actingwhich results in such an acceleration.

Fc = m ac

For objects moving in a circle, Newton’s 2nd Lawtakes the form

F F mamv

rnet c c tan2

What kind of forces might result ina centripetal force?

Force of Tension

What makes this carturn to the right?

Frictional Forceof Tires on Road

Component of Normal Forceof Road on Car

What if the car now goes aroundan inclined bend?

Frictional Forceof Tires on Road

Could point up or downthe road. On what willthe direction of thefrictional force depend?

Angle of the incline

Speed of the car