Question

A ball of radius r and density $\rho$ dropped through a viscous liquid of density $\sigma$ and viscosity $\eta$ attains its terminal velocity at time t, given by $t = A \rho^a r^b \eta^c \sigma^d$, where A is a constant and a, b, c and d are integers. The value of $\frac{b+c{a+d}$ is ___.}

Hint:

The time constant for terminal velocity is mass/damping coefficient ($m/b$). For Stokes flow, $b = 6\pi\eta r$.

Answer 1

Correct Answer: 16

The characteristic time $\tau$ for a particle to reach terminal velocity is given by $\tau = \frac{m}{6\pi \eta r}$.
Mass $m = \frac{4}{3} \pi r^3 \rho$.
Substituting $m$: $t \propto \frac{r^3 \rho}{\eta r} \propto \frac{\rho r^2}{\eta}$.
This gives the dependence: $t = k \rho^1 r^2 \eta^{-1} \sigma^0$.
So, $a=1$, $b=2$, $c=-1$, $d=0$.
We need to find $\frac{b+c}{a+d}$.
$\frac{2 + (-1)}{1 + 0} = \frac{1}{1} = 1$.