Vertical Circular Motion. Roller Coaster Twirling a Ball on a String Vertically x mg T T = 0; v cr mg ma R T mg ma R T cannot be negative, the bob.

Vertical Circular Motion

Roller Coaster

Roller Coaster

Twirling a Ball on a String VerticallyTwirling a Ball on a String Vertically

gRvvgRv

mgR

vmTIfmg

R

vmT

R

vmmgT

R

vmF

topcrtopcr

net

__

22

22

or

0

x

mgT

T = 0; vcr

mgmaR

T

mgmaR T cannot be

negative, the bob will fall down!

mgmaR

If an object is moving in a circular path but at varying speeds, it must have a tangential component to its acceleration as well as the radial one.

Non-uniform Circular Motion

Ferris Wheel

Sustained vertical spinning• Ever wonder what a bike

tire feels like?• At constant speed, the

centripetal acceleration is constant (v2/r), but the direction of gravity keeps changing!

• Feel heavier at bottom than at top

• This ride definitely turns your world around!

Bucket of Water

WE,W

aNB,W

water

Bridges & Speed BumpsBridges & Speed Bumps

gRvN

m

NRgRv

R

vmNmg

R

vmF

Cr

net

0

2

2

The curvature of the bridge/bump and your speed determines if you are going to stay on the bump (N>0) or if you are going to bump: fly off it!

How fast do you need to drive to leave the ground?

Connected BodiesConnected Bodies

Two different masses (m < M) are connected

by a string going through a tube. Predict

what will happen when you whirl the

lighter mass with constant but high speed.

Base your prediction on the CM model.

m

M1. Mass M will be falling down since it is heavier

2. Mass M will be moving up and then stop

3. Mass M will move up and then will fall down

4. It is impossible to whirl the lighter mass around

Connected Bodies - AnalysisConnected Bodies - Analysis

m

MgRv

R

vmMg

TMgR

vmT

FR

vmF

Mnet

mnet

)cos(

)cos(

0

)cos(

0

2

2

_

2

_

If the mv2/R is > Mg, the mass M will accelerate upwards. Notice, the mass m will always rotate so the string is not horizontal. It happens so T has a vertical component to cancel out the mg!

m

M

T

mgx

y

Satellites in Circular Orbits

Weightlessness in Space Shuttle

Artifical Gravity