Centripetal Force Applications

Circus cycle show

The Maximum Speed to Travel in a Horizontal Circular Path

f

f

f

f

f=Fcf=uN

W

N

W=N on a flat,

level surface.

f=μW

μmg= mv2/rμg=v2/r

grv

Maximum speed to round a horizontalcircular path.

gr

2v

Minimum μ toround circular path.

r

An object traveling in a vertical circle:

F

W

Fnet=Fc

At the path bottom:

F-W=mac

F=W+mac

Forces towards the center of thecircular path are positive and forcesaway from the center from the circular path are negative.

WF

At path top:

F+W=mac

F=-W+mac

There is a smaller force applied at thetop of the path than the bottom because gravity aids in the centripetal force on the top of the path. The force has to overcome gravity on the bottom of the path.

The forces on the bottom of a verticalcircular path.

The forces at the top of the circular vertical path.

Minimum speed needed to round a vertical circle:F=0 if there is not enough speedto round the vertical circle 0=W-mac mac=mg

ac=g*v2/r=g

rgv

The Force on a Mass Traveling in a Vertical Circular Path

*The centripetal acceleration must at least be equal to g (9.8 m/s2).

orMinimum speed needed to round a vertical circular path

Vertical Suspension while Traveling a Horizontal Circle

W

f

FcN

Vertical forces

Horizontal forces

ΣFx = Fc

N = Fc

ΣFy = 0f-w=0 W=fmg=μN mg=μ(Fc)mg=μ(mv2/r)g=μv2/r

rg

v

Minimum speed need for an object toremain vertically suspended while traveling around in a horizontalcirclur path.

r

An object spinning horizontally at the end of a string that makes an angle:

W

T

r

Tx

Ty

ΣFx=FcTx=FcTx=mac

Tcosθ=mac

ΣFy=0Ty-W=0Ty=WTsinθ=WTsinθ=mg

Since T is a variable in both equations, then the two equations can be combinedinto one, giving

ca

gtan

θ

Equation Summarygrv

gr

2v

rgv

The maximum speed an object can travel in a circular horizontal path without sliding..The minimum coefficient of friction needed for an object to travel in a horizontal circular path without sliding.

F = W+mac

The force exerted on an object traveling in a vertical circular path at the top.

F = -W+mac

The force exerted on an object traveling in a vertical circular path at the bottom

The minimum speed needed to travel in a vertical circular path.

rg

v The minimum speed needed to remain vertically suspended while traveling in a horizontal circular path.

ca

gtan

A relationship between the angle, acceleration due to gravity, and centripetal acceleration for a mass at the end of a string, rope, or chain that is whirled in a horizontal circular path.