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Question

# A small sphere of radius r falls from rest in a viscous liquid. As a result, heat is produced due to viscous force. The rate of production of heat when the sphere attains its terminal velocity is proportional to:

A
r5
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B
r2
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C
r3
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D
r4
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Solution

## The correct option is A r5A small sphere of radius r falls from rest in a viscous liquid. As a result, heat is produced due to viscous force. The rate of production of heat when the sphere attains its terminal velocity is proportional toLet r is the radius of sphere and vt is its terminal speed. Then the weight of sphere is balanced by the buoyant force and viscous force such that:Weight,w=mg∵ρ=mVm=43πr3ρg........(1)so,w=43πr3ρBuoyant force,FB=43πr3σg..........(2)Where, σ is density of water.Viscous force, F=6πηrvt..............(3)Where, ηis viscosity.From equation (1) (2) and (3)w=FB+Fv43πr3ρg=43πr3σg+6πηrvtvt=29r2(ρ−σ)gη.......(4)The rate of production of heat when the sphere attains its terminal velocity isequal to work done by the viscous forces.W=dQdt=Fv×vtW=6πηrv2tW=6πηr(29r2(ρ−σ)gη)2dQdt∝r5

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