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Question

A cylindrical vessel containing a liquid is rotated about its axis so that the liquid rises at its sides as shown in the figure. The radius of vessel is 5cm and the angular speed of rotation is ωrads-1. The difference in the height, h (in cm) of liquid at the center of the vessel and at the side will be:


A

5ω22g

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B

2ω225g

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C

25ω22g

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D

2ω25g

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Solution

The correct option is C

25ω22g


Step 1. Given data:

Radius of vessel is 5cm

Angular speed of rotation is ωrads-1

Step 2. Solving for height h

Considers a mass m of water, at a distance x from its axis of rotation .The centripetal force will act on liquid fx=mxω2 radially outwards and its weight force fy=mg,

The net work done alongx axis due to centripetal force is,

=0Rm.x.ω2dx=mx2ω220R=mR2ω22

The net work done alongy axis is against the potential energy mgh

Whereh is the height of liquid rising due to rotation andg is acceleration due to gravity

Equating the two equations, we get

mR2ω22=mgh

h=R2ω22g=52ω22g

h=25ω22g

Hence, option C is correct


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