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. $r^3$
• B. $r^2$
• C. $r^5$
• D. $r^4$

Answer 1

The correct option is C $r^5$

Step 1, Given data

The radius of the sphere = r

Let the terminal velocity be $V_t$ .

Step 2,

We know,

Power = Rate of production of heat

Or, $P=F{V}_t$ --------(1)

From the stoke's formula,

$F=6\pi \eta V_{t}r$ ---------------(2)

Substituting equation 2 in equation 1

we get,

$P=6\pi \eta {\left(V_t\right)}^2$ -----------------(3)

We know that the terminal velocity is given as

$V_t=\frac{2r^2(\phi -\sigma )}{9\eta }$ From the above we can say that

$V_t\alpha r^2$ -------------(4)

From the equation 3 and 4 we get

$P\alpha r^5$

Hence the correct option is (D)