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 dHdt is proportional to rn. Find the value of n.
A
5
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B
5.0
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C
5.00
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Solution
Radius 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