Question

A frictionless circular piston of area ${10}^{-2}{m}^2$ and mass 100 kg sinks into a cylindrical container of the same area filled with water of density 1000 $kg/m^3$ as shown in the figure.
The container has a hole of area ${10}^{-3}{m}^2$ at the bottom that is open to the atmosphere. Assuming there is no leakage from the edges of the piston and considering water to be incompressible, the magnitude of the piston velocity (in m/s) at the instant shown is_______(correct to three decimal places).

• A. 1.456

Answer 1

The correct option is A 1.456
$A_1{V}_1=A_2{V}_2$
$V_2=\left(\frac{A_1}{A_2}\right)V_1$
Applying Bernoulli's equation between points 1 and 2, we get
$\frac{p_1}{\rho g}+\frac{V_1^2}{2g}+Z_1=\frac{p_2}{\rho g}+\frac{V_2^2}{2g}+Z_2$


$\left[\frac{p_{atm}+\frac{100\times 10}{{10}^{-2}}}{\rho \times g}\right]+\frac{V_1^2}{2g}+0.5$
$=\frac{p_{atm}}{\rho g}+\frac{A_1^2}{A_2^2}\times \frac{V_1^2}{2g}$
$\frac{100\times 10}{{10}^{-2}\times 1000\times 10}+0.5=(\frac{A_1^2}{A_2^2}-1)\frac{V_1^2}{2g}$
$10+0.5=({10}^2-1)\frac{V_1^2}{2g}$
$V_1^2=\frac{10.5\times 2\times 10}{99}=2.12$
$V_1=1.456m/s$