A Classical Approach of Newtonian Mechanics

7 CIRCULAR MOTION 7.3 Centripetal acceleration

→

'

Y

Q v^

Z

v

v X

P v^

r





Figure 58: Centripetal acceleration.

simply δθ. The vector Z

→

X represents the change in vector velocity, δv, between

times t and t + δt. It can be seen that this vector is directed towards the centre of

the circle. From standard trigonometry, the length of vector ZX is

δv = 2 v sin(δθ/2). (7.12)

However, for small angles sin θ θ, provided that θ is measured in radians.

Hence,

It follows that

δv ' v δθ. (7.13)

δv δθ

a = = v

δt δt

= v ω, (7.14)

where ω = δθ/δt is the angular velocity of the object, measured in radians per

second. In summary, an object executing a circular orbit, radius r, with uniform

tangential velocity v, and uniform angular velocity ω = v/r, possesses an acceler-

ation directed towards the centre of the circle—i.e., a centripetal acceleration—of

magnitude

a = v ω = v^2

r

= r ω^2. (7.15)