First slide

Fluid dynamics

Question

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

Moderate

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

Solution

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
Initial downward force on the flowing out water = Adv²
So equal amount of reaction acts upwards on the cylinder.
$∴$ Initial upward reaction = ${Adv}^{2}$ [As $v = \sqrt{2 gh}$ ]
$∴$ Initial decrease in weight $= Ad (2 gh)$
$= 2 Adgh = 2 \times (\frac{1}{4}) \times 1 \times 980 \times 25 = 12 .5 gm - wt .$

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