Question

When a solid sphere moves through a liquid, the liquid opposes the motion with a force F. The

magnitude of F depends on the coefficient of viscosity $\eta$ of the liquid, the radius r of the sphere and the speed v of the sphere.

Assuming that F is proportional to different powers of these quantities, guess a formula for F using the

method of dimensions. Use F = -$\frac{\eta A(v_2-v_1)}{x_2-x_1}$

• A. $F\alpha \eta rv$
• B. $F\alpha \frac{\eta }{rv}$
• C. $F\alpha {\eta }^2{r}^{\frac{-1}{2}}{v}^3$
• D. None of these

Answer 1

The correct option is A

$F\alpha \eta rv$

Suppose the formula is $F=k{\eta }^a{r}^b{v}^c$

$\eta =\left[M{L}^{-1}{T}^{-1}\right]$

Then, $ML{T}^{-2}=[M{L}^{-1}{T}^{-1}{]}^a{L}^b{\left(\frac{L}{T}\right)}^c$

= $M^a{L}^{-a+b+c}{T}^{-a-c}$

Equating the exponents of M, L and T from both sides,

a = 1

-a + b + c = 1

-a - c = -2

Solving these, a = 1, b = 1, and c = 1

Thus, the formula for F is $F=k\eta rv$