Circular Motion

Lecturer: Professor Stephen T. Thornton

Reading Quiz: When a car is moving around a banked curve, which of the following statements is most true?

A) The friction is always up the slope of the banked road.

B) The friction is always down the slope of the banked road.

C) The friction can either be up or down the slope of the banked road.

D) It sounds as if the car’s tires are too bald to matter.

Answer: C

Friction can be up or down the road. This is in many textbooks and is worked out. There is even one speed where there is no friction.

Last Time

Review

Friction

Today

Circular motion

Centripetal motion, force

Motion on banked curves

Try this with a block of wood. This is similar to the car moving on banked curve.

Conceptual Quiz

Case 1Case 1

Case 2Case 2

Below you see two cases: Below you see two cases: a father a father pullingpulling or or pushingpushing a sled with a force a sled with a force FF that that is applied at an angle is applied at an angle . In . In which case is the normal which case is the normal force greater?force greater?

A) case 1A) case 1B) case 2B) case 2C) it’s the same for bothC) it’s the same for bothD) depends on the magnitude of D) depends on the magnitude of

the force the force FFE) depends on the ice surfaceE) depends on the ice surface

In case 1, the force F is pushing downdown (in addition to in addition to mgmg), so the normal force needs to be largerlarger. In case 2, the force F is pulling upup, against gravity, so the normal force is lessenedlessened.

Conceptual Quiz

Case 1Case 1

Case 2Case 2

Below you see two cases: Below you see two cases: a father a father pullingpulling or or pushingpushing a sled with a force a sled with a force FF that that is applied at an angle is applied at an angle . In . In which case is the normal which case is the normal force greater?force greater?

A) case 1A) case 1B) case 2B) case 2C) it’s the same for bothC) it’s the same for bothD) depends on the magnitude of D) depends on the magnitude of

the force the force FFE) depends on the ice surfaceE) depends on the ice surface

Conceptual Quiz

2m2m mm

FF

Two blocks of masses 2m

and m are in contact on a horizontal frictionless surface. If a force F is applied to mass 2m, what is the force on mass m ? The blocks accelerate.

A) 2A) 2FF

B) B) FF

C) C) FF

D) D) FF

E) E) FF

12

14

13

The force F leads to a specific acceleration of the entire system. In order for mass mass mm to accelerate at the to accelerate at the same rate, the force on it must be same rate, the force on it must be smaller!smaller! How small?? Let’s see...

Conceptual Quiz

2m2m mm

FF

Two blocks of masses 2m and m are in contact on a horizontal frictionless surface. If a force F is applied to mass 2m, what is the force on mass m ? The blocks accelerate.

A) 2A) 2FF

B) B) FF

C) C) FF

D) D) F F

E) E) FF

Follow-up:Follow-up: What is the acceleration of each mass? What is the acceleration of each mass?

12

14

13

F = 3ma; a = F/3m, F1=ma=F/3

Conceptual QuizConceptual QuizA) the force from the rushing air

pushed it off

B) the force of friction pushed it off

C) no net force acted on the box

D) truck went into reverse by accident

E) none of the above

A box sits in a pickup truck

on a frictionless truck bed.

When the truck accelerates

forward, the box slides off

the back of the truck

because:

Generally, the reason that the box in the truck bed would move

with the truck is due to frictionfriction between the box and the bed.

If there is no friction, there is no force to push the box along, If there is no friction, there is no force to push the box along,

and it remains at rest.and it remains at rest. The truck accelerated away, essentially

leaving the box behind!!

Conceptual QuizConceptual QuizA) the force from the rushing air

pushed it off

B) the force of friction pushed it off

C) no net force acted on the box

D) truck went into reverse by accident

E) none of the above

A box sits in a pickup truck

on a frictionless truck bed.

When the truck accelerates

forward, the box slides off

the back of the truck

because:

Mass Moving Up Ramp. A small block of mass m is given an initial speed up a ramp inclined at angle to the horizontal. It travels a distance d up the ramp and comes to rest. (a) Determine a formula for the coefficient of kinetic friction between block and ramp. (b) What can you say about the value of the coefficient of static friction?

q

Circular motion Do demo with

string and ball.

Note that the direction of the velocity is changing. The ball is accelerating!

f iv v v

Notice that tends to point towards the center of the circle. As becomes smaller and smaller, points directly to center. Therefore the acceleration points towards the center of the circle.

v

v

Centripetal accelerationCentripetal means “center seeking”.

Look at your textbook for a derivation of the magnitude of the centripetal acceleration acp :

where r is the radius and v is the speed.

2 1av

v v va

t t

2

cp

va

r

v

a

r

Dynamics of Uniform Circular Motion

We can see that the force must be inward by thinking about a ball on a string. Strings only pull; they never push.

For an object to be in uniform circular motion, there is an acceleration, and, therefore, a net force acting on it.

We already know the acceleration, so we can immediately write the force:

2

R R cp

vF ma ma m

r= = =å

Centripetal forceWhere in the world did this centripetal force come from?

There has to be a force to keep the object moving in a circle. In the case of the ball and string, it is the tension in the string. The tension always points towards the center!

The direction of the centripetal force must also be towards the center!

The moon rotates around the Earth in a circle. What is the centripetal force that causes this?

If you drive around in a circle with a bicycle or even with a car, what is the centripetal force?

In a simple atomic model of the hydrogen atom, the electron rotates around the proton in a circle. What is the centripetal force?

How can we make a bowling ball go around in a circle?

How are you going to do it?

What is the centripetal force?

Circular motionResults for circular motion: Consider an object moving in a circle of

radius r with a constant speed v. A centripetal acceleration of magnitude

v2/r must be present. There must be a centripetal force Fcp of

value 2

cp cp

mvF ma

r

Conceptual Quiz: What other forces are exerted on the ball besides mg?

A) FrictionB) TensionC) A normal force perpendicular to mg.D) A normal force perpendicular to the surface of the cone at the ball.

Answer: D

The only other possible force is the normal force, and it must be perpendicular to the surface that the ball is rolling upon.

N

Conceptual Quiz: What is the direction of the net force?

A) towards the center of the dashed circle at the ball (radially).B) away from the center of the circle at the ball.C) up at the ball.D) down at the ball.E) cannot tell with information given.

Answer: A

Because the ball is moving at constant speed in a circle, the net force must be along the radial direction, towards the center of the circle. This is the centripetal force.

Look at ball moving in vertical circle

In this case we do not have uniform circular motion. The tension always points towards the center, but gravity points down.

Look at Examples in textbook.

Turning corners

What do we notice when we turn corners at high speed?Good chance of falling

over or skidding.When skiing, we lean

over and tilt our skis.Interstate highways

are banked.Motorcycles tilt.

f

Highway Curves: Banked and UnbankedWhen a car goes around a curve, there must be a net force toward the center of the circle of which the curve is an arc. If the road is flat, that force is supplied by friction.

If the frictional force is insufficient, the car will tend to move more nearly in a straight line, as the skid marks show.

As long as the tires do not slip, the friction is static. If the tires do start to slip, the friction is kinetic, which is bad in two ways:

1. The kinetic frictional force is smaller than the static.

2. The static frictional force can point toward the center of the circle, but the kinetic frictional force opposes the direction of motion, making it very difficult to regain control of the car and continue around the curve.

You really do not want your tires to slip!!

Banking the curve can help keep cars from skidding. In fact, for every banked curve, there is one speed at which the entire centripetal force is supplied by the horizontal component of the normal force, and no friction is required. This occurs when:

Whirling Bucket. A bucket of mass 2.00 kg is whirled in a vertical circle of radius 1.10 m. At the lowest point of its motion the tension in the rope supporting the bucket is 25.0 N. (a) Find the speed of the bucket. (b) How fast must the bucket move at the top of the circle so that the rope does not go slack?

Antilock brakes keep the

car wheels from locking

and skidding during a

sudden stop. Why does

this help slow the car

down?

A) k > s so sliding friction is better

B) k > s so static friction is better

C) s > k so sliding friction is better

D) s > k so static friction is better

E) none of the above

Conceptual QuizConceptual Quiz

Antilock brakes keep the

car wheels from locking

and skidding during a

sudden stop. Why does

this help slow the car

down?

A) k > s so sliding friction is better

B) k > s so static friction is better

C) s > k so sliding friction is better

D) s > k so static friction is better

E) none of the above

Static friction is greater than sliding frictionStatic friction is greater than sliding friction, so

by keeping the wheels from skidding, the static

friction force will help slow the car down more

efficiently than the sliding friction that occurs

during a skid.

Conceptual QuizConceptual Quiz

A) component of the gravity force parallel to the plane increased

B) coefficient of static friction decreased

C) normal force exerted by the board decreased

D) both A and C

E) all of A, B, and C

A box sits on a flat board.

You lift one end of the

board, making an angle

with the floor. As you

increase the angle, the

box will eventually begin

to slide down. Why?

Net Force

Normal

Weight

Conceptual QuizConceptual Quiz

A) component of the gravity force parallel to the plane increased

B) coefficient of static friction decreased

C) normal force exerted by the board decreased

D) both A and C

E) all of A, B, and C

A box sits on a flat board.

You lift one end of the

board, making an angle

with the floor. As you

increase the angle, the

box will eventually begin

to slide down. Why?

Net Force

Normal

Weight

As the angle increases, the component of component of

weight parallel to the plane increasesweight parallel to the plane increases and

the component perpendicular to the plane component perpendicular to the plane

decreasesdecreases (and so does the normal force).

Because friction depends on normal force,

we see that the friction force gets smallerfriction force gets smaller

and the force pulling the box down the force pulling the box down the

plane gets biggerplane gets bigger.

Conceptual QuizConceptual Quiz

m

A) not move at all

B) slide a bit, slow down, then stop

C) accelerate down the incline

D) slide down at constant speed

E) slide up at constant speed

A mass m is placed on an

inclined plane ( > 0) and

slides down the plane with

constant speed. If a similar

block (same ) of mass 2m

were placed on the same

incline, it would:

Conceptual QuizConceptual Quiz

The component of gravity acting down

the plane is doubledouble for 2m. However,

the normal force (and hence the friction

force) is also doubledouble (the same factor!).

This means the two forces still cancel

to give a net force of zero.

A mass m is placed on an

inclined plane ( > 0) and

slides down the plane with

constant speed. If a similar

block (same ) of mass 2m

were placed on the same

incline, it would:

W

Nf

Wx

Wy

A) not move at all

B) slide a bit, slow down, then stop

C) accelerate down the incline

D) slide down at constant speed

E) slide up at constant speed

Conceptual QuizConceptual Quiz