Properties of Matter 3 Question 2
8. Water is filled up to a height $h$ in a beaker of radius $R$ as shown in the figure. The density of water is $\rho$, the surface tension of water is $T$ and the atmospheric pressure is $p _0$. Consider a vertical section $A B C D$ of the water column through a diameter of the beaker. The force on water on one side of this section by water on the other side of this section has magnitude
(a) $\left|2 p _0 R h+\pi R^{2} \rho g h-2 R T\right|$
(2007, 3M)
(b) $\left|2 p _0 R h+R \rho g h^{2}-2 R T\right|$
(c) $\left|p _0 \pi R^{2}+R \rho g h^{2}-2 R T\right|$
(d) $\left|p _0 \pi R^{2}+R \rho g h^{2}+2 R T\right|$
Objective Questions II (Only one correct option)
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Answer:
Correct Answer: 8. (a)
Solution:
- Velocity of efflux at a depth $h$ is given by $v=\sqrt{2 g h}$. Volume of water flowing out per second from both the holes are equal.
$$ \begin{aligned} & \therefore \quad a _1 v _1=a _2 v _2 \\ & \text { or } \\ & \left(L^{2}\right) \sqrt{2 g(y)}=\pi R^{2} \sqrt{2 g(4 y)} \text { or } R=\frac{L}{\sqrt{2 \pi}} \end{aligned} $$
