UNIT 2

FORCES

2.1: Uniform Circular Motion

2.2: Inclined Planes

2.3: Statics

Uniform Circular Motion

Chapter 5 Text

Definition

Formulae

Horizontal Circles

Vertical Circles

A ball on a string is swung in a vertical circle. The string

happens to break when it is parallel to the ground and the

ball is moving up. Which trajectory does the ball follow?

1) a

2) b

3) c

4) d

A ball on a string is swung in a vertical circle. The string

happens to break when it is parallel to the ground and the

ball is moving up. Which trajectory does the ball follow?

1) a

2) b

3) c

4) d

Uniform Circular Motion (UCM)

What happens when a non zero net force acts on an object?

It accelerates.

Uniform Circular Motion (UCM)

An object moving in a straight line moves in a straight line if the net force acting on it is: Zero

in which case you have uniform motion,

or Acting in the direction of the motion

in which case you have linear acceleration. The object speeds up, or slows down.

Uniform Circular Motion (UCM)

What happens when the net force acts at an angle to the direction of the motion?

The object moves in a curved path.

Examples:

Projectile motion

A ball twirled on a string

Orbits of planets and moons

Our moon’s orbit is a nearly perfect circle.

So is Earth’s orbit

Definition

An object that moves in a circular path at a constant speed, v, is said to experience uniform circular motion (UCM)

The magnitude of v remains constant (uniform), but the direction of the velocity is continuously changing.

A changing velocity, even if it is direction changing, means accelerated motion.

Which way is the acceleration acting?

The acceleration is acting towards the centre.

This acceleration is called centripetal acceleration, or “centre-seeking” acceleration.

It can also be called radial accelerationsince it is directed along the radius, towards the centre of the circle.

Why is it directed towards the centre?

Recall Newton’s 2nd Law

The acceleration acts in the same direction that the net force acts.

netF ma

What are the forces acting on an object

in a circular path?

It depends.

Consider a ball twirled in a horizontal circle.

What is making the ball go in the circle? The tension in the

string.

With respect to the ball which way is the tension acting? Towards the centre.

What is producing the centripetal forces

in the following situations:

Fc = w = mg

1. A car is rolling over the top of a hill at speed v.

2. Moon orbiting around the Earth.

Fc = Gravity OR1 22c G

GmmF F

r

3. A car traveling on a curve

Fc = Static Friction

Which direction is the Centripetal force acting?

4. A racing car traveling on a banked curve?

Fc = Static Friction + the horizontal component of Gravity caused by the bank (incline)

5. A ball twirled in a vertical circle

Fc depends on the position of the ball.

What is making the ballgo in a circle at the top of the path?

Gravity and Tension

5. A ball twirled in a vertical circle

What is making the ballgo in a circle at thebottom of the path?

Tension

In which case, top or bottomwould the tension be the greatest?

At the Bottom

Practice:

Page 247 #1, 2

Lab 2 Page 245

Formulae for UCM

Centripetal acceleration

where

v is the constant speed of the object

r is the radius of the circular path

2

c

va

r

What does the following do to the

centripetal acceleration?

Increase the object’s speed

This would greatly increase the object’s acceleration

If you double the speed, the ac would increase by a factor of

Increase the radius of the path.

This would decrease the object’s acceleration.

If you double the radius, the ac would decrease by a factor of

four

a half

Formulae for UCM

Centripetal acceleration

To find the objects speed, v:

2

c

va

r

dv

t

What is d?

d is the distance around The circle

Consider one revolution:

d = circumference of circle

t = T = period Period is the time to make one complete

revolution

2d r

2d rv

t T

Substitute this into:

Centripetal acceleration

2

c

va

r

22

c

rTar

2 2

2

4 1c

ra

T r

2

2

4c

ra

T

Recall 1f

T

22 2

2

44c

ra rf

T

Newton’s 2nd Law applied to UCM

netF ma

net c cF F ma

2

c

vF m

r

Examples of Horizontal Circles

Orbits

1. The moon has a nearly perfect orbit about the earth with a period of 27.3 days. The radius of the orbit is 385 000 km.

A) Determine the acceleration of the moon towards the earth.

Examples of Horizontal Circles

Orbits

B) Calculate the centripetal force acting on the moon. The moon has a mass of 7.36 x 1022 kg.

C) What causes this force?

Gravity

Examples of Horizontal Circles

2. Ball on a String (easy one)

A 150 g ball at the end of a string is swinging in a horizontal circle on a frictionless table. The radius is 0.60 m. The ball makes exactly 2 revolutions in a second. What is the tension in the string?

Examples of Horizontal Circles

3. Ball on a String (not so easy one)

A 2.0 kg ball suspended by a string is traveling in a horizontal circular path. If the tension in the string is 52 N and the radius of the circle is 1.5 m, what is the speed of the ball?

Examples of Horizontal Circles

4. Turntable

A fly has a coefficient of static friction of 0.10 between itself and a phonograph record of the BeeGees. If the fly begins at the centre and walks straight towards the edge how far will it get before it slides off? The record is spinning at 33 rpm and the fly has a mass of 0.12 g

Examples of Horizontal Circles

5. Car going around a turn.

What is the maximum speed at which a 1250 kg car can round a flat, curved road, without slipping, if the radius of the curve is 88.0 m and the coefficient of static friction is 0.50?

Examples of Horizontal Circles

6. Amusement ride

In the "Rotor Ride" at an amusement park, riders are pressed against the inside vertical wall of a rotating drum 2.5 m in radius that has an angular speed of 27 rev/min about a vertical axis. Viewed from a stationary frame of reference outside the drum, what are the magnitude and direction of the force of the wall on the rider whose mass is 70 kg? The rider's weight is borne by the floor, while the drum is rotating, the floor is removed. If the coefficient of static friction between the rider and the wall is 0.6, will he slip or will he be held pressed against the vertical wall?

Examples of Horizontal Circles

Amusement ride

Examples:

Page 206

#1, 2, 3, 4, 5, 6

Vertical Circles

1. Suppose we have a ball swung on a string in a vertical circle. The ball is swung at a speed of 12 m/s.

The mass of the ball is 0.25 kg.

The string is 1.25 m long.

Find the tension in thestring at top and bottom of the swing.

Top of Swing

What is causing Fc here?

Tension and Gravity2

net c g

vF F F T m

r

2

g

vT m F

r

2

2

(12 )0.25 0.25 9.8

1.25

mmsT kg kg

m s

Bottom of Swing

What is causing Fc here?

Tension

(Gravity is acting against it)

2

g

vT m F

r

2

2

(12 )0.25 0.25 9.8

1.25

mmsT kg kg

m s

2

net c g

vF F T F m

r

2.The biker in the picture on the previous slide fell from his bike at the very top of the loop. If the loop has a radius of 2.3 m, what would be the minimum speed he would need to complete the loop safely?

What provides the centripetal force at the top

of the loop?

However at the minimum speed the Normal Force can be reduced to near zero.

Gravity and Normal Force

2

net c g

vF F F m

r

2gF r mv

gF rv

m

mgrv

m

v gr 2

9.8 2.3m

v ms

2

net c g N

vF F F F m

r

3. A car goes over the top of a hill that has a radius of 275 m. If the has a mass of 1500 kg, what is the maximum speed that car can go over top without becoming airborne? (Express the answer in km/h)

What provides the centripetal force at the top of the hill?

Gravity (the Normal Force acts against Fc)

However at the maximum speed the

Normal Force will be reduced to near zero.2

net c g

vF F F m

r

2

net c g N

vF F F F m

r

2

g

vF mg m

r

2vmg m

r

2gr v

v gr 29.8 275msv m

June 2006 Public

A pail of water on the end of a string revolves at a uniform rate in a vertical circle of radius 85.0 cm. Its speed is 4.15 m/s and the mass of the pail and water together is 1.00 kg.

2% (i) Calculate the magnitude of the tension in the string when the pail is at the top of its path.

Top of Swing

2

net c g

vF F F T m

r

2

g

vT m F

r

2

2

(4.15 )1.00 1.00 9.8

0.85

mmsT kg kg

m s

2% (ii) At what minimum speed must the pail be

travelling when upside down at the top of the circle

so that the water does not fall out?

At the minimum speed tension approaches zero.

2

net c g

vF F F T m

r

2

g

vF m

r

v gr

29.8 0.85msv m

Chief Markers Report

During a roller coaster ride the riders move through two loops, the second being one half the radius of the first. The riders travel at the same speed at the top of each of these two loops.

Using principles of physics, explain why riders would experience a greater normal force at the top of the second, smaller loop than at the top of the first, larger loop.

4%

Chief Markers Report

Page 214

#1, 2, 3

Page 226

#16, 17, 20, 21, 22, 23

Banking of Curves

Curves on highways are banked, computed to be safe at some particular speed. The normal force exerted by the roadbed has a horizontal component (not along the incline plane) that serves as the centripetal force.

This means no friction (or less friction) is required to safely go around the turn

The result is that a car can negotiate a properly banked curve even on glare ice. (ms = 0)

For these problems we assume that the car does not move up or down the bank

Thus for the Vertical Forces

0netyF

0Ny gF F

Ny gF F

cosN gF F …1

Horizontal Forces

Solve Eq 1 for FN

netx cF F

Nx cF F2

N

vF sin m

r …2

cosN gF F

cosg

N

FF

cosN

mgF

Sub into eq 2

This shows that the angle of the banking depends on the speed and on the radiusof the curve. The mass (or size) of the vehicle does not

matter

cosN

mgF

2

N

vF sin m

r

2

cos

mg vsin m

r

2

cos

sin mvmrg

2

tanvrg

Example

Determine the bank angle of the a highway where the posted max speed is 90 km/h if the radius of the curve is 242 m

2

tanvrg

1 90 /3.6tan

242 9.8

kms

msm

If a car attempts to go around the turn at a speed greater than 90 km/h, what is required to keep the car going in the circle?

Friction

In this case the friction prevents the car from going up the bank thus making a larger radius turn.

If a car attempts to go around the turn at a speed less than 90 km/h, friction is required to keep it from sliding down the bank making a smaller radius curve

Page 214

#4

Page 226

#18, 19

Centrifugal Force

http://www.physicsclassroom.com/mmedia/circmot/rcd.html

http://dev.physicslab.org/Document.aspx?doctype=3&filename=CircularMotion_CentripetalForceExamples.xml

http://www.physics.umanitoba.ca/~birchall/PHYS1020/LectureNotes/Lecture15.pdf

http://www.sparknotes.com/physics/dynamics/uniformcircularmotion/problems.html