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 spheres P and Q, we will use the formula for terminal velocity \( V_t \) of a sphere falling through a fluid, which is given by:

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

where:
- \( r \) = radius of the sphere
- \( \sigma \) = density of the sphere
- \( \rho \) = density of the fluid
- \( g \) = acceleration due to gravity (we can take it as constant for both spheres)
- \( \eta \) = viscosity of the fluid

### Step 1: Calculate the radius of each sphere
- For sphere P (diameter = 1 cm):
\[ r_P = \frac{1 \text{ cm}}{2} = 0.5 \text{ cm} \]

- For sphere Q (diameter = 0.5 cm):
\[ r_Q = \frac{0.5 \text{ cm}}{2} = 0.25 \text{ cm} \]

### Step 2: Identify the densities and viscosities
- Density of both spheres \( \sigma = 8 \text{ gm/cm}^3 \)
- Density of liquid for sphere P \( \rho_P = 0.8 \text{ gm/cm}^3 \)
- Density of liquid for sphere Q \( \rho_Q = 1.6 \text{ gm/cm}^3 \)
- Viscosity of liquid for sphere P \( \eta_P = 3 \text{ poise} \)
- Viscosity of liquid for sphere Q \( \eta_Q = 2 \text{ poise} \)

### Step 3: Write the formula for terminal velocities
- For sphere P:
\[ V_P = \frac{2r_P^2 (\sigma - \rho_P) g}{9 \eta_P} \]

- For sphere Q:
\[ V_Q = \frac{2r_Q^2 (\sigma - \rho_Q) g}{9 \eta_Q} \]

### Step 4: Calculate the terminal velocities
- Substitute the values for sphere P:
\[ V_P = \frac{2(0.5)^2 (8 - 0.8) g}{9 \times 3} \]
\[ = \frac{2 \times 0.25 \times 7.2 g}{27} \]
\[ = \frac{3.6 g}{27} \]

- Substitute the values for sphere Q:
\[ V_Q = \frac{2(0.25)^2 (8 - 1.6) g}{9 \times 2} \]
\[ = \frac{2 \times 0.0625 \times 6.4 g}{18} \]
\[ = \frac{0.8 g}{18} \]

### Step 5: Find the ratio of terminal velocities
Now, we can find the ratio \( \frac{V_P}{V_Q} \):
\[ \frac{V_P}{V_Q} = \frac{\frac{3.6 g}{27}}{\frac{0.8 g}{18}} \]

### Step 6: Simplify the ratio
\[ = \frac{3.6 \times 18}{0.8 \times 27} \]
\[ = \frac{64.8}{21.6} \]
\[ = 3 \]

### Final Answer
The ratio of the terminal velocities of spheres P and Q is:
\[ \frac{V_P}{V_Q} = 3 \]