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 relationship between mass, density, and volume
The mass of a planet can be expressed in terms of its density and volume. The formula for mass \( M \) is:
\[ M = \rho \times V \]
where \( \rho \) is the density and \( V \) is the volume.

### Step 2: Calculate the volume of the planets
The volume \( V \) of a sphere (which we assume the planets are) is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Thus, for two planets with radii \( r_1 \) and \( r_2 \):
- For Planet 1:
\[ M_1 = \rho \times \frac{4}{3} \pi r_1^3 \]
- For Planet 2:
\[ M_2 = \rho \times \frac{4}{3} \pi r_2^3 \]

### Step 3: Write the formula for acceleration due to gravity
The acceleration due to gravity \( g \) at the surface of a planet is given by:
\[ g = \frac{G M}{r^2} \]
where \( G \) is the gravitational constant.

### Step 4: Substitute the mass expressions into the gravity formula
For Planet 1:
\[ g_1 = \frac{G M_1}{r_1^2} = \frac{G \left(\rho \times \frac{4}{3} \pi r_1^3\right)}{r_1^2} = \frac{4}{3} \pi G \rho r_1 \]
For Planet 2:
\[ g_2 = \frac{G M_2}{r_2^2} = \frac{G \left(\rho \times \frac{4}{3} \pi r_2^3\right)}{r_2^2} = \frac{4}{3} \pi G \rho r_2 \]

### Step 5: Compare the two accelerations
Now we can compare \( g_1 \) and \( g_2 \):
\[ \frac{g_1}{g_2} = \frac{\frac{4}{3} \pi G \rho r_1}{\frac{4}{3} \pi G \rho r_2} = \frac{r_1}{r_2} \]
This shows that the ratio of the accelerations due to gravity is directly proportional to the ratio of their radii.

### Step 6: Conclusion
Since \( r_1 \) and \( r_2 \) are different, it follows that:
- If \( r_1 > r_2 \), then \( g_1 > g_2 \) (the larger planet has greater acceleration due to gravity).
- If \( r_1 < r_2 \), then \( g_1 < g_2 \) (the smaller planet has lesser acceleration due to gravity).

Thus, the acceleration due to gravity is greater on the planet with the larger radius.

### Final Answer
The acceleration due to gravity is greater on the larger planet.

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