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

r^{3}

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r^{2}

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r^{5}

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r^{4}

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Solution

The correct option is C $r^{5}$
Rate of heat produced
$\frac{d Q}{d t} = F_{v} \times V_{T} (∵ V_{T} = \frac{2 r^{2}}{9 η} (ρ - σ) g) = 6 π η r V_{T} \times V_{T} (∵ V_{T} \propto r^{2}) \frac{d Q}{d t} \propto r {V_{T}}_{T}^{2} (∵ V_{T} \propto r^{2}) ∴ \frac{d Q}{d t} \propto r^{5}$

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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

The correct option is C r5Rate of heat produced dQdt=Fv×VT (∵VT=2r29η(ρ−σ)g)=6πηrVT×VT (∵VT∝r2)dQdt∝rVT 2 (∵VT∝r2)∴ dQdt∝r5

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

The correct option is C $r^{5}$
Rate of heat produced
$\frac{d Q}{d t} = F_{v} \times V_{T} (∵ V_{T} = \frac{2 r^{2}}{9 η} (ρ - σ) g) = 6 π η r V_{T} \times V_{T} (∵ V_{T} \propto r^{2}) \frac{d Q}{d t} \propto r {V_{T}}_{T}^{2} (∵ V_{T} \propto r^{2}) ∴ \frac{d Q}{d t} \propto r^{5}$