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A small sphere falls from rest in a visc...

A small sphere falls from rest in a viscous liquid. Due to friction, heat is produced. Find the relation between the rate of production of heat and the radius of the sphere at terminal velocity.

Text Solution

Verified by Experts

The correct Answer is:
`(dQ)/(dt)propr^(5)`
Terminal velocity `v_(T) = (2r^(2)g)/(9eta)(rho_(s) - rho_(L))`
and viscous force `F = 6pirv_(T)`
Viscous force is the dissipative force. Hence.
`(dQ)/(dt) = Fv_(T) = (6pietarv_(T))(v_(T)) = 6pietarv_(T)^(2)`
`= 6pietar{(2)/(9)(r^(2)g)/(eta)(rho_(s) - rho_(L))}^(2) = (8pig^(2))/(27eta)(rho_(s) - rho_(L))^(2)r^(5) = (dQ)/(dt)propr^(5)`
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