Question

A liquid drop of density $\rho$, radius $r$, and surface tension $\sigma$ oscillates with time period T. Which of the following expressions for $T^2$ is correct?

• A. $\frac{\rho r^3}{\sigma }$
• B. $\frac{\rho \sigma }{r^3}$
• C. $\frac{r^3\sigma }{\rho }$
• D. None of these

Answer 1

Answer: (A)

$\frac{\rho r^3}{\sigma }$

Let $T\rho r\sigma$

Dimersim of $T=\left[T\right]$ $\rho$ is density

Dimersim of $r=\left[L\right]$ $r$ is radius

Dimersim of $\sigma =\left[m{T}^{-2}\right]$ $\sigma$ is surface Tension

Dimersim of $\rho =\left[m{L}^{-3}\right]$ Then $T^2=?$

$T=r^a{\sigma }^b{\rho }^c$

$\left[T\right]=\left[L{]}^a\right[m{T}^{-2}{]}^b[m{L}^{-3}{]}^c$

comparative on both sides

$a-3c=0$ $12-2b$

$b+c=0$

Solving $=b=-1/2,c=1/2a=3/2$

$\therefore T=r^{3/2}{\sigma }^{-1/2}{\rho }^{1/2}$

$T=\sqrt{\frac{r^3\rho }{\sigma }}$

$T^2=\frac{r^3\rho }{\sigma }$ $AnswerisA$