Two planets have same density but different radii The acceleration due to gravity would be .

Options

A. Same on both planets
B. Greater on the smaller planet
C. Greater on the larger planet
D. Dependent on the distance of planet from the sun

Answer

To solve the problem of comparing the acceleration due to gravity on two planets with the same density but different radii, 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 universal gravitational constant, \( M \) is the mass of the planet, and \( R \) is the radius of the planet.

### Step 2: Express the mass in terms of density and volume
The mass \( M \) of a planet can be expressed in terms of its density \( \rho \) and volume \( V \):
\[ M = \rho \cdot V \]
For a spherical planet, the volume \( V \) is given by:
\[ V = \frac{4}{3} \pi R^3 \]
Thus, we can write:
\[ M = \rho \cdot \frac{4}{3} \pi R^3 \]

### Step 3: Substitute the mass into the gravity formula
Substituting the expression for mass into the formula for \( g \):
\[ g = \frac{G \cdot \left(\rho \cdot \frac{4}{3} \pi R^3\right)}{R^2} \]
This simplifies to:
\[ g = \frac{4}{3} \pi G \rho R \]

### Step 4: Analyze the relationship between \( g \) and \( R \)
From the equation \( g = \frac{4}{3} \pi G \rho R \), we can see that the acceleration due to gravity \( g \) is directly proportional to the radius \( R \) of the planet when the density \( \rho \) is constant.

### Step 5: Compare the two planets
Let’s denote the two planets as Planet 1 and Planet 2, with radii \( R_1 \) and \( R_2 \) respectively. Since both planets have the same density \( \rho \):
\[ g_1 = \frac{4}{3} \pi G \rho R_1 \]
\[ g_2 = \frac{4}{3} \pi G \rho R_2 \]
Now, taking the ratio of \( g_1 \) to \( g_2 \):
\[ \frac{g_1}{g_2} = \frac{R_1}{R_2} \]

### Conclusion
From the ratio, we can conclude:
- If \( R_1 < R_2 \), then \( g_1 < g_2 \) (the smaller planet has a smaller acceleration due to gravity).
- If \( R_1 > R_2 \), then \( g_1 > g_2 \) (the larger planet has a greater acceleration due to gravity).

Thus, the acceleration due to gravity is not the same; it depends on the radius of the planets.