Two solid spheres of same metal but of mass M and 8M fall simultaneously on a viscous liquid and their terminal velocitied are v and n v, then value of n is

Options

A. 16
B. 8
C. 4
D. 2

Explanation

To solve the problem, we need to analyze the relationship between the masses of the two spheres, their radii, and their terminal velocities as they fall through a viscous liquid.

### Step 1: Understand the relationship between mass and volume
The mass \( m \) of a sphere can be expressed in terms of its volume \( V \) and density \( \rho \):
\[ m = V \cdot \rho \]
For a solid sphere, the volume \( V \) is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Thus, we can express mass in terms of radius:
\[ m = \frac{4}{3} \pi r^3 \cdot \rho \]

### Step 2: Relate the masses and radii of the spheres
Let the mass of the first sphere be \( M \) and the mass of the second sphere be \( 8M \). Using the relationship derived above:
1. For the first sphere:
\[ M = \frac{4}{3} \pi r_1^3 \cdot \rho \]
2. For the second sphere:
\[ 8M = \frac{4}{3} \pi r_2^3 \cdot \rho \]

### Step 3: Find the relationship between the radii
From the equations for mass, we can express the radii in terms of mass:
\[ r_1^3 = \frac{3M}{4\pi \rho} \]
\[ r_2^3 = \frac{3 \cdot 8M}{4\pi \rho} = 6M \]
Taking the cube root gives us:
\[ r_1 = \left(\frac{3M}{4\pi \rho}\right)^{1/3} \]
\[ r_2 = \left(\frac{24M}{4\pi \rho}\right)^{1/3} = 2 \left(\frac{3M}{4\pi \rho}\right)^{1/3} = 2r_1 \]

### Step 4: Relate terminal velocities to radius
The terminal velocity \( v_t \) of a sphere falling through a viscous liquid is given by:
\[ v_t \propto r^2 \]
Thus, we can write:
\[ v_{t1} \propto r_1^2 \]
\[ v_{t2} \propto r_2^2 \]

### Step 5: Calculate the ratio of terminal velocities
Using the relationship \( r_2 = 2r_1 \):
\[ v_{t2} \propto (2r_1)^2 = 4r_1^2 \]
Now, we can write the ratio of terminal velocities:
\[ \frac{v_{t1}}{v_{t2}} = \frac{r_1^2}{4r_1^2} = \frac{1}{4} \]
This implies:
\[ \frac{v}{nv} = \frac{1}{4} \]
From this, we can conclude:
\[ n = 4 \]

### Final Answer
The value of \( n \) is \( 4 \).

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