A small steel ball of radius r is allowed to fall under gravity through a column of a viscous liquid of coefficient of viscosity. After some time the velocity of the ball attains a constant value known as terminal velocity vₜ. The terminal velocity depends on (i) the mass of the ball. (ii) η (iii) r and (iv) acceleration due to gravity g. which of the following relations is dimensionally correct

The correct relation is:
$$v_t\propto \frac{mg}{\eta r}$$

To determine which relation is dimensionally correct, we first establish the dimensional formulas for each of the physical quantities involved:

1. Terminal Velocity ($v_t$):
Velocity is displacement per unit time, so its dimensions are:
$$\left[v_t\right]=\left[M^{0}L{T}^{-1}\right]$$

2. Mass ($m$):
The dimensional formula for mass is:
$$\left[m\right]=\left[M\right]$$

3. Acceleration due to gravity ($g$):
Since it is an acceleration, its dimensions are:
$$\left[g\right]=\left[L{T}^{-2}\right]$$

4. Radius ($r$):
Radius represents a length, so its dimensions are:
$$\left[r\right]=\left[L\right]$$

5. Coefficient of viscosity ($\eta$):
Using viscous force formula, $F=6\pi \eta rv$, we can express $\eta$ as:
$$\eta =$$constant × F r v
Substituting the dimensions of force $\left[ML{T}^{-2}\right]$, length $\left[L\right]$, and velocity $\left[L{T}^{-1}\right]$:
$$\left[\eta \right]=\frac{\left[ML{T}^{-2}\right]}{\left[L\right]\left[L{T}^{-1}\right]}=\left[M{L}^{-1}{T}^{-1}\right]$$

Now, we find the dimensions of the expression $\frac{mg}{\eta r}$:
$$\left[\frac{mg}{\eta r}\right]=\frac{\left[M\right]\left[L{T}^{-2}\right]}{\left[M{L}^{-1}{T}^{-1}\right]\left[L\right]}$$

Simplifying the denominator:
$$\left[\eta \right]\left[r\right]=\left[M{L}^{-1}{T}^{-1}\right]\left[L\right]=\left[M{T}^{-1}\right]$$

Substituting this back into the relation:
$$\left[\frac{mg}{\eta r}\right]=\frac{\left[ML{T}^{-2}\right]}{\left[M{T}^{-1}\right]}=\left[L{T}^{-1}\right]$$

Comparing the results, the dimensions of the right-hand side, $\left[L{T}^{-1}\right]$, exactly match the dimensions of the terminal velocity $\left[v_t\right]$.
Therefore, the relation $v_t\propto \frac{mg}{\eta r}$ is dimensionally correct.