A time exposure photo of a rolling disk shows that a point on the edge traces out a curve called a “cycloid”.

As the wheel rolls to the right, points O and P move forward a distance, s. During this same time, a point on the wheel’s rim has moved through an arc length, ____.s

Rs definition of arc length…

take derivative of both sides…

Condition for rolling… ONLY true if wheel “rolls” smoothly and does NOT slip/skid.... NOTE: This is different from v = R for pure rotation. Taking the derivative one more time…

Rdt

d

dt

ds

Rvcm Racm

Rolling can be viewed as simply the combination of Rotation about a fixed point (O) and Translation …

Pure Rotation

Pure Translation Rolling

It’s crazy, but when a wheel is rolling, the speed at the top point is _____, while the bottom point is _________! Look back at the picture of the rolling disk on 1st slide…

2vcm

-vcm+ vcm= 0

2vcm stationary

If you’re having a hard time “buying” it, then check out this picture of a rolling wheel… It’s blurry on top, where it is moving the fastest!

Also, think about the motion of a tank… The tread in contact with the ground is stationary!! DEMO with toy tank… Demo: Wheel and Strobe Light

So, in this circus act, who must move faster, the horse or the dog?

The ______, who must also walk __________!!

dog

backwards

Since the bottom point (the point in contact with the ground) is stationary, rolling can be alternatively looked at as “Pure Rotation” about that point. The velocity of the cm and of point T can then be calculated from v = R where R is the distance from the rotational axis…..

cmv

Tv

cm

R)2( R cmv2

cm

Kinetic Energy, K, of a Rolling Object:

If we take the view of rolling as “pure rotation” about point P (the rotational axis), then….

2

2

1 PIK Where IP = moment of inertia of object about point P (need parallel axis theorem to find it)

22

2

1 MRIK cm

22

2

1

2

1cmcm mvIK

Rotational KE Translational KE

222

2

1

2

1 mRIcm

The “Role” of Friction in RollingWithout _________, a wheel cannot roll.

• If a wheel rolls without slipping/sliding, the frictional force is _________ friction.

•If a wheel slips (skids), the frictional force is __________ friction.

Application: Anti-Lock Brake Systems (ABS)A car can only be steered when static friction is present. So ABS senses when the wheels are sliding (Vcm = r) and then automatically “pumps” the brakes (> 10 times per sec) so that the wheels will roll, allowing the driver to keep control of the car. Also, for most surfaces, static > kinetic, so required stopping distance is less if rolling. (Gravel is an exception.)

friction

static

kinetic

Examples:

1.If the surface of an incline is frictionless, a wheel placed on the incline would ________ down the ramp, but NOT ______, because there is NO ________ about the center of mass.

mg

From an energy perspective…. If the wheel is released from a height, h….

mgh

translate roll

torque

2

2

1cmmv

2. If there IS friction present, then Fs (static friction) opposes the wheel’s tendency to slide down due to gravity. _______ is then created about the center of mass and the wheel _____.

mg

Fs

From an energy perspective…. If the wheel is released from a height, h….

Will the Vcm be faster or slower if the wheel rolls compared to if it just translated down the ramp? Vcm at bottom will be _________ if it rolls.

Torquerolls

mgh 22

2

1

2

1 cmcm Imv

slower

To analyze rolling using Newton’s 2nd Law, again, translation AND rotation must be considered…. mg

Fs

maF I

And, finally, one more equation will be necessary to relate the linear and angular acceleration…

mgsin

cms maFmg sin IRFs

Racm

R

• If the wheel is just on the verge of sliding, then Fs = Fs,max = N.

• If the surfaces are too slick or if the angle is too large, the frictional force becomes _________ friction, and the wheel _______. Some of the initial energy is lost to ______.

3. What happens when a rolling wheel encounters a frictionless incline?

The wheel will ___________________, because there is no friction to produce a torque to angularly decelerate the wheel. Even at its highest point it will still be ________.

kinetic slidesheat

NOT stop rotating

spinning

4. What is direction of the frictional force that acts on an accelerating bicycle?

From an energy perspective….

222

2

1

2

1

2

1 cmcmcm ImghImv

As the bicyclist pedals faster, the wheel rotates faster (increase in ). This increase tends to slide the wheel towards the _____. The frictional force, as always, opposes the impending motion, and acts towards the ______, supplying the external force on the bicycle that makes it accelerate.

left

right

Why doesn’t a wheel that is “rolling without slipping” on a horizontal surface roll forever since the frictional force is static, producing no energy loss?

No object “perfectly rolls”…. All rolling objects deform to some extent, producing what is called “rolling friction”.

Perfect Rolling – One contact point Realistic Rolling

Properly inflated tires improve a car’s gas mileage. Why does this make sense?