Question

A body of mass `m` hung at one end of the spring executes simple harmonic motion . The force constant of a spring is `k` while its period of vibration is `T`. Prove by dimensional method that the equation `T = 2 pi m // k` is correct. Dervive the correct equation , assuming that they are related by a power law.

Answer 1

Correct Answer - ` C sqrt((m)/(k))`
The given equation is `T = ( 2 pi m)/( k)`
Taking the dimensions of both sides, we have
`[T] = ([M])/( [ ML^(0)T^(-2)] = T^(2)`
As the dimensions of two sides are not equal , hence the equation is incorrect.
Let the correct relation be `T = Cm^(a) k^(b) , where C` is constant. Equating the dimensions of both sides , we get
`[T] = [M]^(a) [MT^(-2)]^(b)`
or `[M^(0) L^(0) T] = [M^(a+b)L^(0)T^(-2b)]`
Comparing the powers of M,L, and T on both sides , we get ` a + b = 0 and -2b = 1`.
Therefore , `b = -(1)/( 2) and a = (1)/(2)`
:. `T = C m^(-1//2) k ^(-1//2) = C sqrt((m)/(k))`
This is the correct equation.