Two planets have radii `r_(1) and r_(2)` and densities `d_(1) and d_(2)` respectively. Then the ratio of acceleration due to gravity on them is

Options

A. `r_(1) d_(1) : r_(2) d_(2)`
B. `r_(1) d_(2) : r_(2) d_(1)`
C. `r_(1)^(2) d_(1) : r_(2)^(2) d_(2)`
D. `r_(2)^(2) d_(1) : r_(1)^(2) d_(2)`

Explanation

To find the ratio of acceleration due to gravity on two planets with given radii and densities, we can follow these steps:

### Step 1: Understand the formula for acceleration due to gravity
The acceleration due to gravity \( g \) on the surface of a planet is given by the formula:
\[ g = \frac{G \cdot M}{r^2} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the planet, and \( r \) is the radius of the planet.

### Step 2: Express the mass of the planets in terms of density
The mass \( M \) of a planet can be expressed in terms of its density \( d \) and volume \( V \):
\[ M = d \cdot V \]
The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Thus, the mass of planet 1 (with radius \( r_1 \) and density \( d_1 \)) is:
\[ M_1 = d_1 \cdot \left(\frac{4}{3} \pi r_1^3\right) \]
And for planet 2 (with radius \( r_2 \) and density \( d_2 \)):
\[ M_2 = d_2 \cdot \left(\frac{4}{3} \pi r_2^3\right) \]

### Step 3: Substitute the mass into the gravity formula
Now, substituting the mass expressions into the gravity formula for both planets:
For planet 1:
\[ g_1 = \frac{G \cdot M_1}{r_1^2} = \frac{G \cdot \left(d_1 \cdot \frac{4}{3} \pi r_1^3\right)}{r_1^2} = \frac{G \cdot d_1 \cdot \frac{4}{3} \pi r_1^3}{r_1^2} = \frac{G \cdot d_1 \cdot \frac{4}{3} \pi r_1}{1} \]

For planet 2:
\[ g_2 = \frac{G \cdot M_2}{r_2^2} = \frac{G \cdot \left(d_2 \cdot \frac{4}{3} \pi r_2^3\right)}{r_2^2} = \frac{G \cdot d_2 \cdot \frac{4}{3} \pi r_2^3}{r_2^2} = \frac{G \cdot d_2 \cdot \frac{4}{3} \pi r_2}{1} \]

### Step 4: Find the ratio of \( g_1 \) to \( g_2 \)
Now, we can find the ratio of the accelerations due to gravity:
\[ \frac{g_1}{g_2} = \frac{\frac{G \cdot d_1 \cdot \frac{4}{3} \pi r_1}{1}}{\frac{G \cdot d_2 \cdot \frac{4}{3} \pi r_2}{1}} \]
The \( G \) and \( \frac{4}{3} \pi \) terms cancel out:
\[ \frac{g_1}{g_2} = \frac{d_1 \cdot r_1}{d_2 \cdot r_2} \]

### Step 5: Write the final result
Thus, the ratio of the acceleration due to gravity on the two planets is:
\[ g_1 : g_2 = d_1 r_1 : d_2 r_2 \]

### Final Answer
The ratio of acceleration due to gravity on the two planets is \( \frac{g_1}{g_2} = \frac{d_1 r_1}{d_2 r_2} \).