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Chapter 6

Circular Motion

and

Other Applications of Newton’s

Laws

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Uniform Circular Motion

Uniform circular motion occurs when an objectmoves in a circular path with a constant speed

The associated analysis motion is a particle in 

uniform circular motion   An acceleration exists since the direction of the

motion is changing

This change in velocity is related to an acceleration

The velocity vector is always tangent to the path ofthe object

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Changing Velocity in Uniform

Circular Motion

The change in thevelocity vector is due to

the change in direction

The vector diagramshows  f i  v v v

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Centripetal Acceleration, cont

The magnitude of the centripetal acceleration vectoris given by

The direction of the centripetal acceleration vector isalways changing, to stay directed toward the centerof the circle of motion

2

C 

va

r 

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Period

The period , T , is the time required for onecomplete revolution

The speed of the particle would be thecircumference of the circle of motion dividedby the period

Therefore, the period is defined as

2 r T 

v 

 

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Tangential Acceleration

The magnitude of the velocity could also be changing

In this case, there would be a tangential acceleration 

The motion would be under the influence of both

tangential and centripetal accelerations Note the changing acceleration vectors

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Total Acceleration

The tangential acceleration causes thechange in the speed of the particle

The radial acceleration comes from a changein the direction of the velocity vector

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Total Acceleration, equations

The tangential acceleration:

The radial acceleration:

The total acceleration:

Magnitude

Direction

Same as velocity vector if v is increasing, opposite if v isdecreasing

t 

dv a 

dt 

2

r C 

v a a 

r 

2 2

r t a a a 

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Uniform Circular Motion, Force

A force, , isassociated with thecentripetal acceleration

The force is alsodirected toward thecenter of the circle

Applying Newton’s

Second Law along theradial direction gives

2

c 

v F ma m  

r 

r F

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Uniform Circular Motion, cont

A force causing a centripetalacceleration acts toward thecenter of the circle

It causes a change in thedirection of the velocity vector

If the force vanishes, theobject would move in a

straight-line path tangent tothe circle

See various release points inthe active figure

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Motion in a Horizontal Circle

The speed at which the object movesdepends on the mass of the object and thetension in the cord

The centripetal force is supplied by thetension

Tr 

v  m 

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Conical Pendulum

The object is inequilibrium in thevertical direction andundergoes uniformcircular motion in thehorizontal direction ∑Fy = 0 → T cos θ = mg

∑Fx = T sin θ = m ac

v is independent of m 

sin tanv Lg   

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Horizontal (Flat) Curve

The force of static frictionsupplies the centripetalforce

The maximum speed atwhich the car can negotiatethe curve is

Note, this does not dependon the mass of the car

s v gr  

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Banked Curve

These are designed withfriction equaling zero

There is a component ofthe normal force thatsupplies the centripetalforce

tan v rg 

 

2

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Banked Curve, 2

The banking angle is independent of themass of the vehicle

If the car rounds the curve at less than thedesign speed, friction is necessary to keep itfrom sliding down the bank

If the car rounds the curve at more than the

design speed, friction is necessary to keep itfrom sliding up the bank

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Loop-the-Loop

This is an example of avertical circle

At the bottom of the

loop (b), the upwardforce (the normal)experienced by theobject is greater than

its weight 2

2

1

bot 

bot 

mv F n mg  

r 

v n mg 

rg 

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Loop-the-Loop, Part 2

At the top of the circle(c), the force exertedon the object is less

than its weight2

2

1

top 

top 

mv F n mg  

r 

v 

n mg  rg 

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Non-Uniform Circular Motion

The acceleration andforce have tangentialcomponents

produces thecentripetalacceleration

  produces thetangentialacceleration

r F

t F

r t  F F F

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Vertical Circle with Non-

Uniform Speed

The gravitational forceexerts a tangentialforce on the object

Look at the componentsof Fg 

The tension at anypoint can be found

2

cosv 

T mg Rg 

 

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Top and Bottom of Circle

The tension at the bottom is a maximum

The tension at the top is a minimum

If T top = 0, then topv gR 

2

1bot v T mg 

Rg 

2

1top v 

T mg Rg