Two spherical planets have the same mass but densities in the ratio 1:8. For these planets,the

a) acceleration due to gravity will be in the ratio 4:1
b) acceleration due to gravity will be in the ratio 1:4
c) escape velocities from their surfaces will be in the ratio `sqrt2:1`
d) escape velocities from their surfaces will be in the ratio `1: sqrt2`

Options

A. only b&d are true
B. only b&c are true
C. only a,c&d are true
D. only a&b are true

Answer

To solve the problem, we need to find the ratios of the acceleration due to gravity and escape velocities of two spherical planets that have the same mass but different densities.

### Given:
- The ratio of densities \( \rho_1 : \rho_2 = 1 : 8 \)
- Both planets have the same mass \( M \)

### Step 1: Relate Density to Volume and Radius
The mass of a planet can be expressed in terms of its density and volume:
\[ M = \rho \cdot V \]
For a sphere, the volume \( V \) is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Thus, we can write:
\[ M = \rho \cdot \frac{4}{3} \pi r^3 \]
For both planets, we have:
\[ M_1 = \rho_1 \cdot \frac{4}{3} \pi r_1^3 \]
\[ M_2 = \rho_2 \cdot \frac{4}{3} \pi r_2^3 \]
Since \( M_1 = M_2 \), we can equate the two expressions:
\[ \rho_1 \cdot r_1^3 = \rho_2 \cdot r_2^3 \]

### Step 2: Substitute the Density Ratio
Given the density ratio \( \frac{\rho_1}{\rho_2} = \frac{1}{8} \), we can express \( \rho_2 \) in terms of \( \rho_1 \):
\[ \rho_2 = 8 \rho_1 \]
Substituting this into the mass equation gives:
\[ \rho_1 \cdot r_1^3 = 8 \rho_1 \cdot r_2^3 \]
Dividing both sides by \( \rho_1 \) (assuming \( \rho_1 \neq 0 \)):
\[ r_1^3 = 8 r_2^3 \]

### Step 3: Find the Ratio of Radii
Taking the cube root of both sides:
\[ \frac{r_1}{r_2} = \sqrt[3]{8} = 2 \]
Thus, the ratio of the radii is:
\[ r_1 : r_2 = 2 : 1 \]

### Step 4: Calculate Acceleration Due to Gravity
The acceleration due to gravity \( g \) on the surface of a planet is given by:
\[ g = \frac{GM}{r^2} \]
For both planets, we have:
\[ g_1 = \frac{GM}{r_1^2} \]
\[ g_2 = \frac{GM}{r_2^2} \]
Now, taking the ratio:
\[ \frac{g_1}{g_2} = \frac{r_2^2}{r_1^2} \]
Substituting \( r_1 = 2r_2 \):
\[ \frac{g_1}{g_2} = \frac{r_2^2}{(2r_2)^2} = \frac{r_2^2}{4r_2^2} = \frac{1}{4} \]
Thus, the ratio of the acceleration due to gravity is:
\[ g_1 : g_2 = 1 : 4 \]

### Step 5: Calculate Escape Velocity
The escape velocity \( v \) from the surface of a planet is given by:
\[ v = \sqrt{2g r} \]
For both planets:
\[ v_1 = \sqrt{2g_1 r_1} \]
\[ v_2 = \sqrt{2g_2 r_2} \]
Taking the ratio:
\[ \frac{v_1}{v_2} = \frac{\sqrt{g_1 r_1}}{\sqrt{g_2 r_2}} = \sqrt{\frac{g_1}{g_2} \cdot \frac{r_1}{r_2}} \]
Substituting \( \frac{g_1}{g_2} = \frac{1}{4} \) and \( \frac{r_1}{r_2} = 2 \):
\[ \frac{v_1}{v_2} = \sqrt{\frac{1}{4} \cdot 2} = \sqrt{\frac{2}{4}} = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} \]
Thus, the ratio of escape velocities is:
\[ v_1 : v_2 = 1 : \sqrt{2} \]

### Final Answers:
- The acceleration due to gravity will be in the ratio \( 1 : 4 \) (Option b).
- The escape velocities from their surfaces will be in the ratio \( 1 : \sqrt{2} \) (Option d).