Question

Given M is the mass suspended from a spring of force constant k. The dimensional formula for ${[M/k]^{1/2}}$ is same as that for:
A.) Frequency
B.) Time period
C.) Velocity
D.) Wavelength

Answer 1

Start by writing the dimensional formula of all the given quantities. Then try to find the dimensional formula for k using Hooke's law. Substitute this value of k in the given relation to find out the final dimensional formula, compare this dimensional formula with different known physical quantities which suits the best.

Complete answer:
Given M is mass
K is spring constant
Let us write the dimensional formula of each of given quantity
So, $M = [M]$
For the dimensional formula of k , we would need to derive it.
We know, from Hooke's law $F = - kx$, where F is the restoring force, x is the displacement from the mean position.
$\Rightarrow k = \dfrac{F}{x}$
Substituting the dimensional formula of force $F = [ML{T^{ - 2}}]$ and displacement$x = [L]$, we get
$k = \dfrac{{[ML{T^{ - 2}}]}}{{[L]}} \\\ \Rightarrow k = [M{T^{ - 2}}] \\\$
Now , we need to find out dimensional formula of ${[M/k]^{1/2}}$, therefore substituting the values we get
${\left( {\dfrac{M}{k}} \right)^{1/2}} = {\left( {\dfrac{{[M]}}{{[M{T^{ - 2}}]}}} \right)^{1/2}} \\\ \Rightarrow {\left( {\dfrac{M}{k}} \right)^{1/2}} = {\left( {\dfrac{1}{{{T^{ - 2}}}}} \right)^{1/2}} \\\ \Rightarrow {\left( {\dfrac{M}{k}} \right)^{1/2}} = T \\\$
Which is nothing but the time or time period.

So, the correct answer is “Option B”.

Note:
Dimensional analysis and formulas are very useful in cross verifying the results or formulas obtained , so if at any instance we feel the formula seems to be slightly unbalanced , we can use this. Also , basic dimensional formulas must be known or must be on the fingertips to make the process faster such as force, acceleration, velocity etc.