Suppose we are told that the acceleration a of a particle moving
with uniform speed v in a circle of radius r is proportional
to some power of r, say rn, and some power of v, say vm. How
can we determine the values of n and m?

Question

Suppose we are told that the acceleration a of a particle moving
with uniform speed v in a circle of radius r is proportional
to some power of r, say rn, and some power of v, say vm. How
can we determine the values of n and m?

anonymous · Answer

by dimensional analysis..

anonymous · Answer

Acceleration is measured in m/s^2...so we need to get these units out of v^mr^n.  Velocity is measured in m/s and r=distance is measure in meters.  The only way to get seconds^2 out of this is if m=2.  Now our units become (m^2/s^2)r^n.  We need to get rid of one power of our distance unit.  Thus n=-1.  This gives us the correct dimensions for acceleration (m/s^2).

anonymous · Answer

uniform velocity means the paticle only centripital acceleration
$$mv ^{2}/r=k m(v ^{^{n/2}})^{2}/r ^{m}$$
on copmring.
n=2,m=-1