Two planets of radii `r_1` and `r_2` are made from the same material. The ratio of the acceleration due to gravities at the surface of the two planets is

Options

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

Answer

To find the ratio of the acceleration due to gravity at the surfaces of two planets made from the same material, we can follow these steps:

### Step 1: Understand the formula for acceleration due to gravity
The acceleration due to gravity \( g \) at the surface of a planet is given by the formula:
\[ g = \frac{GM}{r^2} \]
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 mass in terms of density and volume
Since the planets are made from the same material, they have the same density \( \rho \). The mass \( M \) of a planet can be expressed in terms of its density and volume:
\[ M = \rho V \]
The volume \( V \) of a sphere (which we assume the planets are) is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Thus, we can write:
\[ M = \rho \left(\frac{4}{3} \pi r^3\right) \]

### Step 3: Substitute mass into the formula for \( g \)
Substituting the expression for mass into the formula for \( g \):
\[ g = \frac{G \left(\rho \frac{4}{3} \pi r^3\right)}{r^2} \]
This simplifies to:
\[ g = \frac{4}{3} \pi G \rho r \]
Here, we see that \( g \) is directly proportional to the radius \( r \) of the planet when the density \( \rho \) is constant.

### Step 4: Calculate the ratio of gravitational accelerations
Let \( g_1 \) be the acceleration due to gravity on planet 1 with radius \( r_1 \), and \( g_2 \) be the acceleration due to gravity on planet 2 with radius \( r_2 \):
\[ g_1 = \frac{4}{3} \pi G \rho r_1 \]
\[ g_2 = \frac{4}{3} \pi G \rho r_2 \]
Now, we can find the ratio of the two accelerations:
\[ \frac{g_1}{g_2} = \frac{\frac{4}{3} \pi G \rho r_1}{\frac{4}{3} \pi G \rho r_2} \]
The constants \( \frac{4}{3} \pi G \rho \) cancel out, leading to:
\[ \frac{g_1}{g_2} = \frac{r_1}{r_2} \]

### Final Answer
Thus, the ratio of the acceleration due to gravity at the surfaces of the two planets is:
\[ \frac{g_1}{g_2} = \frac{r_1}{r_2} \]

---