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. Determine the values of n and m and write the simplest form of an equation for the acceleration.

Answer 1

the value of m and n can be obtained using a tool called dimensional analysis
acceleration a has dimension\[L T^{-2}\] the velocity has dimension \[LT ^{-1}\] and radius of a circle (or any path) is L, the equation of acceleration, can be written \[a = v^{m}r^{n}\]. put the dimensions of a, v, r in the equation respectively yields
\[LT ^{-2}=\left( LT ^{-1} \right)^{m}\left( L \right)^{n}\]. Simplyfing the results, \[LT ^{-2}=L^{m+n}T^{-m}\]. By comparing the power of L and T on the left hand with the right hand one, we have\[m=2\] and \[m+n=1\]. Substitung m=2 into the last equation,we obtain n=-1.And finally we have equation of a in terms of v and r by substituting m and n\[a=v^{m}r ^{n}\] \[a=v ^{2}r ^{-1}\] or \[a=\frac{v ^{2} }{ r }\] which is the wellknown centripetal acceleration

Answer 2

thank you