The force F acting on a body moving in a circular path depends on mass of the body (m) velocity(v) and radius (r) of the circular path. Obtain the expression for the force by dimensional analysis method `(k=1)`

Explanation

The principle of homogeneity of dimensions states that the dimensions of all the terms in a physical expression should be the same. For example, in the physical expression `v^2 = u^2 + 2as` , the dimensions of `v^2 , u^2` and 2 as are the same and equal to `[L^2 T^(-2) ]`
` F prop m^a v^b r^c`
`F = K m^z v^b r^c ` K = 1
` F = m^a v^b r^c`
Dimensionally `[MLT^(-2) ] = [M]^a [LT^(-1) ]^b [L]^c`
Campare the power of M,L and T
a = 1 ...(1)
b + c = 1 ....(2)
- b = - 2
b = 2 ....(3)
sub (3) and (2) we get
b+c=1
2 + c = 1
c = -1 .....(4)
Substitute a, b and c value in force equation
`F = m^a v^b r^c`
` = mv^2 r^(-1)`
`F = (mv^2)/(r )`
This equation is known as centripetal force.