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Question

# A small spherical body of radius r is falling under gravity in a viscous medium and due to friction, the medium gets heated. When the body attains terminal velocity, then the rate of heating is proportional to:

A
r
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B
r3/2
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C
r5
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D
r1/2
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Solution

## The correct option is C r5Radius of the spherical body =r Rate of heat production=Power dissipated & Power dissipated=F×v ...(i) Hence, rate of heat production is, dHdt=(6πrη)v2 [∵F=6πrηv is the viscous force] ⇒dHdt=(6πηr)[29(σ−ρ)r2gη]2 ...(ii) σ & ρ are the density of solid body and liquid respectively. v=vT, when terminal velocity is achieved. From Eq.(ii), rate of heat production, =dHdt∝r5 Hence, the correct option is (c)

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