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Ch5Circular Motion and Force

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Centripetal Force - Swinging Ball• Any body rotating about a fixed point will experience a centripetal (center

seeking) acceleration.• This acceleration is always directed inwards towards the center of the

circle.• If asked to find the centripetal force, then simply multiply the acceleration

by the mass.• When viewing “Swinging Ball” problems, note that the body has a force

(tension, normal,…) acting on it that is pulling the body towards the center of the circle at all times.

• In this problem that force is the tension in the string.

a

2va

r

v

2vF m

r

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• You are looking down on a flail spinning in a horizontal circle. • The chain breaks on the flail.• Which flail (red, blue, or green) travels in the path that the flail

would follow after the chain broke?• Explain your reasons for selecting the flail you picked.

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Friction Types for Rolling and Skidding Tires

• What type of friction acts between the road and a tire that is rolling without skidding?

• What type of friction acts between the road and a tire that is skidding?

• Watch the red dot on the tire below as it rolls and then slides.

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Skidding and Curves• When an ATV safely negotiates a curve, the

centripetal acceleration produced by the inward acting static friction is able to accelerate (change the direction of) the ATV.

• The tracks left behind by this ATV would look like the ones to the right.

• When an ATV goes around the curve at too high a speed, the tires begin to skid.

• The tracks left behind by this ATV would look like the ones to the right.

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Negotiating Curves• Watch the animation to the right and

explain what has happened.

• What force acted on the ATV that prevented it from slipping off the road during the first lap and in which direction was it acting?

• The static friction directed inwards towards the center of curvature.

• What forces were acting on the ATV when it slipped off the road?

• Excess speed caused the tires to skid and to slide changing the static friction to a much smaller kinetic friction that was not able to make the ATV change direction.

sF

ca

W

N

kF

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Negotiating Curves - Unbanked• When a car negotiates an unbanked

curve, the only force causing the centripetal acceleration is the static friction acting inwards along the radius of curvature.

• This static friction accelerates the car and causes it to turn.

• Use Newton’s laws to derive an expression for the maximum speed the ATV can have to safely negotiate a curve of radius r when the coefficient of static friction between the tires and the road is s.

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x FS

vF F m

r

sF

ca

W

N

0yF N W

FS sF N

N W mg

FS sF mg

2

s

vmg m

r

sv gr

FS sF N

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Centripetal Force - Swinging Ball• We will derive the equations relating the tensions in the top and bottom of

a vertically swinging ball to the velocities of the ball at the top and bottom of its swing.

• Notice that the tension is pulling the ball towards the center of rotation at all times.

• Which tension would you expect to be the largest: top or bottom? Why?• Derive top:• Derive bottom.• Do our equations agree with our speculations?

a

v

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Centripetal Force – Ferris Wheel• As the man riding the Ferris

wheel is moving in a circle, he experiences a centripetal force (red) at all points as shown.

• The seat is also exerting an upward normal force on him (blue).

.

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Centripetal Force – Ferris Wheel • We start with Newton’s second law and proceed as below (top in left

column bottom in right column).• Think about the results.• Which normal force would you expect to be the largest: top or bottom?

Why?• Derive top:• Derive bottom:• Do our equations agree with our speculations?

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Circular Motion and Force• Synchronized jet fighters

perform a perfect vertical circular loop maneuver.

• The jets are going 1622 km/h at the bottom of the loop.

• What is the minimum diameter of the loop needed in order to prevent the pilot from experiencing no more than 5.5 g (5.5 times the normal value of Earth’s gravity)?

• What is the apparent weight of the pilot at the top and bottom of the loop?

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Centripetal Force – Ferris Wheel• Often you will be asked to calculate the maximum

speed that a body may have in order to remain in the Ferris Wheel without being buckled in.

• What do you think the value of the normal force would be at the top of the circle if the body was traveling at this maximum speed?

• N = 0.

• Derive max v

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Circular Motion and Force• A test dummy (m = 62.5 kg) rides freely in a high-speed roller coaster with

a radius of curvature of r = 52.00 m.• What type of a problem is this one?• A Ferris wheel problem.

• If the coaster was moving at vT = 18.00 m/s at the top of the coaster, then what is the apparent weight of the test dummy at this point?

• What is the maximum safe speed of the roller coaster?

RUN

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Circular Motion and Force• An airplane, whose engine failed, was gliding to the ground.• Once the engine restarted, the pilot “pulled up” in order to keep the plane

from crashing into the ground.• The radius of curvature of the pull up is 175.0 m, and the plane’s speed at

the lowest point of this curve is 78.0 m/s.• What is the apparent weight of the 58.2 kg pilot at this lowest point in

flight?