Question

The period of oscillation '$T$' of a loaded spring depends upon the mass of load '$M$' and force constant $K$ of the spring. If the constant of proportionality is $2\pi$, then the dimensional formula for '$T$' is:

• A. $T=2\pi \frac{M}{K}$
• B. $T=2\pi \frac{K}{M}$
• C. $T=2\pi \sqrt{\frac{K}{M}}$
• D. $T=2\pi \sqrt{\frac{M}{K}}$

Answer 1

The correct option is D $T=2\pi \sqrt{\frac{M}{K}}$

$T=M^{\alpha }{K}^{\beta }$

$M^0{L}^0{T}^1=M^{\alpha }(M^1{T}^{-2}{)}^{\beta }$

$M^0{L}^0{T}^1=M^{\alpha +\beta }{T}^{-2\beta }$
$\alpha +\beta =0$
$-2\beta =1$ $\Rightarrow \beta =-\frac{1}{2},\alpha =\frac{1}{2}$
$\Rightarrow T\alpha \sqrt{\frac{M}{K}}$ $\Rightarrow T=2\pi \sqrt{\frac{M}{K}}$