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

Options

A. `v_(T) prop (mg)/(eta r)`
B. `v_(T) prop (etar)/(mg)`
C. `v_(T) prop eta r mg`
D. `v_(T) prop (mgr)/(eta)`

Answer

To solve the problem of determining which relation for terminal velocity \( \upsilon_T \) is dimensionally correct, we will analyze the dependencies on mass \( m \), viscosity \( \eta \), radius \( r \), and acceleration due to gravity \( g \).

### Step 1: Identify the dimensions of the variables involved

1. **Mass \( m \)**: The dimension of mass is \( [M] \).
2. **Viscosity \( \eta \)**: The dimension of viscosity can be derived from the equation of viscous force:
\[ F = \eta A \frac{dv}{dx} \]
- Force \( F \): \( [F] = [M][L][T^{-2}] = [MLT^{-2}] \)
- Area \( A \): \( [A] = [L^2] \)
- Change in velocity \( dv \): \( [dv] = [LT^{-1}] \)
- Change in distance \( dx \): \( [dx] = [L] \)

Rearranging gives:
\[ \eta = \frac{F}{A \frac{dv}{dx}} = \frac{[MLT^{-2}]}{[L^2] \cdot [LT^{-1}]/[L]} = [ML^{-1}T^{-1}] \]

3. **Radius \( r \)**: The dimension of radius is \( [L] \).
4. **Acceleration due to gravity \( g \)**: The dimension of acceleration is \( [g] = [LT^{-2}] \).

### Step 2: Determine the dimension of terminal velocity \( \upsilon_T \)

The dimension of terminal velocity is:
\[ [\upsilon_T] = [LT^{-1}] \]

### Step 3: Analyze each option for dimensional correctness

1. **Option A**: \( \upsilon_T \propto \frac{mg}{\eta r} \)
- Dimensions:
\[ [mg] = [M][LT^{-2}] = [MLT^{-2}] \]
\[ [\eta] = [ML^{-1}T^{-1}] \]
\[ [r] = [L] \]
- Therefore:
\[ \frac{mg}{\eta r} = \frac{[MLT^{-2}]}{[ML^{-1}T^{-1}][L]} = \frac{[MLT^{-2}]}{[MLT^{-1}]} = [LT^{-1}] \]
- This matches the dimension of velocity \( [LT^{-1}] \). **Option A is correct**.

2. **Option B**: \( \upsilon_T \propto \frac{1}{\frac{mg}{\eta r}} \)
- This would give dimensions of \( [LT^{-1}]^{-1} \), which does not match \( [LT^{-1}] \). **Option B is incorrect**.

3. **Option C**: \( \upsilon_T \propto \eta r mg \)
- Dimensions:
\[ [\eta r mg] = [ML^{-1}T^{-1}][L][MLT^{-2}] = [M^2L^2T^{-3}] \]
- This does not match \( [LT^{-1}] \). **Option C is incorrect**.

4. **Option D**: \( \upsilon_T \propto \frac{mg}{\eta r^2} \)
- Dimensions:
\[ \frac{mg}{\eta r^2} = \frac{[MLT^{-2}]}{[ML^{-1}T^{-1}][L^2]} = \frac{[MLT^{-2}]}{[ML^{-1}T^{-1}]} = [LT^{-1}] \]
- This matches the dimension of velocity \( [LT^{-1}] \). **Option D is also correct**.

### Conclusion

The dimensionally correct relations for terminal velocity \( \upsilon_T \) are found in **Option A** and **Option D**.