The time period of oscillation of a body is given by $T=2 \pi \sqrt{\frac{m g A}{K}}$, $K$ represent

Question

The time period of oscillation of a body is given by $T=2 \pi \sqrt{\frac{m g A}{K}}$, $K$ represents the kinetic energy, $m$ mass, $g$ acceleration due to gravity and $A$ is unknown. If $[A]=\mathrm{M}^{x} \mathrm{~L}^{y} \mathrm{~T}^{z}$; then what is the value of $x+y+z$ ?\

Professor PC Thomas Classes · Answer

\ $ \begin{aligned} & 	ext {  (a): }[T]=\left[\left(\frac{m g A}{K}ight)^{1 / 2}ight]=\left[\frac{m g A}{\left(m v^2ight)}ight]^{1 / 2}=\left[\frac{A}{v T}ight]^{1 / 2}=\left[\frac{A}{L}ight]^{1 / 2} \ & \Rightarrow \quad[A]=\left[\mathrm{M}^0 \mathrm{LT}^2ight] \Rightarrow\left[\mathrm{M}^x \mathrm{~L}^y \mathrm{~T}^zight]=\left[\mathrm{M}^0 \mathrm{LT}^2ight] \ & \Rightarrow \quad x=0, y=1, z=2 \Rightarrow x+y+z=3 \end{aligned} $