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Formula for centripetal acceleration and force

Consider an object of mass m moving with constant angular velocity (ω) and constant speed (v) in a circle of radius r with centre O (see Figure 1).
It moves from P to Q in a time t.
Change in velocity parallel to PO = vsinθ - 0
Change in velocity perpendicular to PO = vcosθ - v
When θ becomes small (that is when Q is very close to P) sinθ is close to θ in radians and cosq tends to 1.

The equations then become:
Change in velocity along PO = vθ - 0 = vθ
Change in velocity perpendicular to PO = v - v = 0
Therefore acceleration along PO = vθ/t = vω = v2/r = ω2r




Centripetal acceleration (a) = v2/r = ω2r

Applying Newton's Second Law (F = ma) gives:


Centripetal force (F) = mv2/r = mω2r


schoolphysics: Circular motion animation

To see an animation of velocity and force in circular motion please click on the animation link.



schoolphysics: Circular motion - varying speed animation

To see an animation of cicular motion with varying speed please click on the animation link.



Example problems
1. A space station has a radius of 100 m and is rotated with an angular velocity of 0.3 radians per second.
(i) which side of a "room" at the rim is the floor
(ii) what is the artificial gravity produced at the rim

(i) the floor is the outer rim of the space station
(ii) a = g = π2r = 0.32x100 = 9 ms-2

2. Calculate the rate of rotation for a space station of radius 65 m so that astronauts at the outer edge experience artificial gravity equal to 9.8 ms2.

But T = 2πr/v and so T = 16.18 s giving the rotation rate (1/T) as 0.062 Hz.
 

A VERSION IN WORD IS AVAILABLE ON THE SCHOOLPHYSICS USB
 
 
 
 
© Keith Gibbs 2020