The velocity of small spherical ball of mass ' m ' and density ' d 1 ', when dropped in a container filled with glycerine becomes constant after some time. The viscous force acti

Question

The velocity of small spherical ball of mass ' m ' and density ' d 1 ​ ', when dropped in a container filled with glycerine becomes constant after some time. The viscous force acting on the ball if density of glycerine is ' d 2 ​ ' is

TestHub · Accepted Answer

When the velocity of the ball becomes constant, it has achieved terminal velocity. At this stage, the net force acting on the ball is zero.

The forces acting on the ball are:
Downward weight of the ball, $W = m g$
Upward buoyant force, $U = V d_{2} g$
Upward viscous force, $F_{v}$

Equating the forces:
$W = U + F_{v}$
$F_{v} = W - U$

Since mass $m = V d_{1}$ , the volume of the ball is $V = \frac{m}{d _{1}}$ .

Substituting $V$ in the expression for buoyant force:
$U = (\frac{m}{d _{1}}) d_{2} g = m g (\frac{d _{2}}{d _{1}})$

Therefore, the viscous force is:
$F_{v} = m g - m g (\frac{d _{2}}{d _{1}}) = m g (1 - \frac{d _{2}}{d _{1}})$

Answer: $mg (1 - \frac{d _{2}}{d _{1}})$

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