Question

Check the correctness of formula, T=2ᴫ√m/k , T be the time period, m be the mass and k be
the force per unit displacement

Answer 1

To check:

Correctness of the formula :

$\boxed{ \sf{T = 2\pi \sqrt{ \dfrac{m}{k} } }}$

Calculation:

We shall use DIMENSIONAL ANALYSIS to check the correctness of the above equation. In this analysis , we shall try to check the dimensions on the LHS and RHS :

LHS:

$\therefore \: \bigg \{T \bigg \} = \bigg \{time \bigg\}$

RHS:

$\therefore \: \bigg \{2\pi \sqrt{ \dfrac{m}{k} } \bigg \} \equiv \bigg \{ \sqrt{ \dfrac{m}{k} } \bigg \}$

$\implies \: \bigg \{2\pi \sqrt{ \dfrac{m}{k} } \bigg \} \equiv \bigg \{ \sqrt{ \dfrac{m}{ (\frac{force}{x} )} } \bigg \}$

$\implies \: \bigg \{2\pi \sqrt{ \dfrac{m}{k} } \bigg \} \equiv \bigg \{ \sqrt{ \dfrac{M}{ (\frac{ML{T}^{ - 2} }{L} )} } \bigg \}$

$\implies \: \bigg \{2\pi \sqrt{ \dfrac{m}{k} } \bigg \} \equiv \bigg \{ \sqrt{ \dfrac{1}{ {T}^{ - 2} } } \bigg \}$

$\implies \: \bigg \{2\pi \sqrt{ \dfrac{m}{k} } \bigg \} \equiv \bigg \{ \sqrt{ {T}^{2} } \bigg \}$

$\implies \: \bigg \{2\pi \sqrt{ \dfrac{m}{k} } \bigg \} \equiv \bigg \{ T \bigg \}$

$\implies \: \bigg \{2\pi \sqrt{ \dfrac{m}{k} } \bigg \} \equiv \bigg \{ time \bigg \}$

So, LHS = RHS .

Hence, the equation is dimensionally correct.

[Hence checked]