Consider two solid spheres P and Q each of density `8gm cm^-3` and diameters 1cm and 0.5cm, respectively. Sphere P is dropped into a liquid of density `0.8gm cm^-3` and viscosity `eta=3` poiseulles. Sphere Q is dropped into a liquid of density `1.6gmcm^-3` and viscosity `eta=2` poiseulles. The ratio of the terminal velocities of P and Q is

Explanation

To find the ratio of the terminal velocities of two solid spheres P and Q, we will use the formula for terminal velocity \( V_t \) of a sphere falling through a fluid:

\[ V_t = \frac{2R^2(\sigma - \rho)g}{9\eta} \]

where:
- \( R \) = radius of the sphere
- \( \sigma \) = density of the sphere
- \( \rho \) = density of the fluid
- \( g \) = acceleration due to gravity (constant)
- \( \eta \) = viscosity of the fluid

### Step-by-Step Solution

1. **Identify the parameters for both spheres:**
- For sphere P:
- Diameter = 1 cm → Radius \( R_P = \frac{1}{2} \) cm = 0.5 cm
- Density \( \sigma_P = 8 \) gm/cm³
- Density of liquid \( \rho_1 = 0.8 \) gm/cm³
- Viscosity \( \eta_1 = 3 \) poise

- For sphere Q:
- Diameter = 0.5 cm → Radius \( R_Q = \frac{0.5}{2} \) cm = 0.25 cm
- Density \( \sigma_Q = 8 \) gm/cm³
- Density of liquid \( \rho_2 = 1.6 \) gm/cm³
- Viscosity \( \eta_2 = 2 \) poise

2. **Write the formula for terminal velocity for both spheres:**
- For sphere P:
\[ V_{tP} = \frac{2R_P^2(\sigma_P - \rho_1)g}{9\eta_1} \]
- For sphere Q:
\[ V_{tQ} = \frac{2R_Q^2(\sigma_Q - \rho_2)g}{9\eta_2} \]

3. **Calculate the terminal velocity ratio \( \frac{V_{tP}}{V_{tQ}} \):**
\[ \frac{V_{tP}}{V_{tQ}} = \frac{R_P^2(\sigma_P - \rho_1)}{\eta_1} \cdot \frac{\eta_2}{R_Q^2(\sigma_Q - \rho_2)} \]

4. **Substituting the values:**
- \( R_P = 0.5 \) cm, \( R_Q = 0.25 \) cm
- \( \sigma_P = \sigma_Q = 8 \) gm/cm³
- \( \rho_1 = 0.8 \) gm/cm³, \( \rho_2 = 1.6 \) gm/cm³
- \( \eta_1 = 3 \) poise, \( \eta_2 = 2 \) poise

5. **Calculate the differences in densities:**
- For sphere P: \( \sigma_P - \rho_1 = 8 - 0.8 = 7.2 \) gm/cm³
- For sphere Q: \( \sigma_Q - \rho_2 = 8 - 1.6 = 6.4 \) gm/cm³

6. **Now substituting into the ratio:**
\[ \frac{V_{tP}}{V_{tQ}} = \frac{(0.5)^2 \cdot 7.2}{3} \cdot \frac{2}{(0.25)^2 \cdot 6.4} \]
\[ = \frac{0.25 \cdot 7.2}{3} \cdot \frac{2}{0.0625 \cdot 6.4} \]
\[ = \frac{1.8}{3} \cdot \frac{2}{0.4} \]
\[ = 0.6 \cdot 5 = 3 \]

### Final Answer:
The ratio of the terminal velocities of spheres P and Q is \( \frac{V_{tP}}{V_{tQ}} = 3 \).