Question

In your reading on two dimensional kinematics, you should have seen that the acceleration of a particle experiencing uniform circular motion is given by:

a=v^2/r

where v is the speed of the particle and r is the radius of the circular path it is following. What I would like for you to do is a qualitative analysis to help you understand and explain physically why the acceleration has this form. (Please draw images to understand)

(a) Compare two particles travelling at the same speed, but around circular paths of different radii (say, one of them twice the radius of the other). Construct the instantaneous acceleration vector on each circle.

(b) Compare two particles travelling around circles of the same radius but one of them faster than the other (say, twice as fast). Construct the instantaneous acceleration vector.

(c) Use your diagrams to explain physically why this equation for the acceleration depends on v and r the way that it does.

Answer 1

(a) First figure showing the two circle one of radius R and other of radius 2R. Therefore the circle of the higher radius will have less acceleration.
(b) In the second figure the two circle are of same radius but the speed is different therefore the particle have the higher speed will have higher acceleration.
(c) From the newton second law
ma = mV²/R
a = V²/R

Radial acceleration is toward centre of the circle