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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.
 6 months ago
 6 months 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.
 6 months ago
 6 months ago

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adekurniawanBest ResponseYou've already chosen the best response.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
 6 months ago
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