## saysaban 2 years ago 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.

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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

2. saysaban

thank you