Question

A cylinder containing water up to a height of 25 cm has a hole of cross-section $\frac { 1 } { 4 } \mathrm {~cm} ^ { 2 }$ in its bottom. It is counterpoised in a balance. What is the initial change in the balancing weight when water begins to flow out

• A. Increase of 12.5 gm-wt
• B. Increase of 6.25 gm-wt
• C. Decrease of 12.5 gm-wt
• D. Decrease of 6.25 gm-wt

Answer 1

Answer: (C)

Decrease of 12.5 gm-wt

Answer 2

Let A = The area of cross section of the hole, v

= Initial velocity of efflux, d = Density of water,

Initial volume of water flowing out per second = Av

Initial mass of water flowing out per second = Avd

Rate of change of momentum = Adv²

$\therefore$ Initial downward force on the out flowing water = Adv²

So equal amount of reaction acts upwards on the cylinder.

∴ Initial upward reaction =$A d v ^ { 2 }$

[As $v = \sqrt { 2 g h }$]

$\therefore$ Initial decrease in weight $= 2 \mathrm { Adgh }$

$= 2 \times \left( \frac { 1 } { 4 } \right) \times 1 \times 980 \times 25 = 12.5$ gm-wt