For moon, its mass is 1/81 of earth mass and its diameter is 1/3.7 of earth dia. If acceleration due to gravity at earth surface is 9.8 `m//s^2` then at moon its value is :

Options

A. 2.86 `m//s^2`
B. 1.65 `m//s^2`
C. 8.65 `m//s^2`
D. 5.16 `m//s^2`

Explanation

To find the acceleration due to gravity on the Moon, we can use the relationship between the gravitational acceleration on two different bodies (Earth and Moon) based on their masses and radii. The formula for gravitational acceleration is given by:

\[ g = \frac{GM}{R^2} \]

Where:
- \( g \) is the acceleration due to gravity,
- \( G \) is the universal gravitational constant,
- \( M \) is the mass of the body,
- \( R \) is the radius of the body.

### Step 1: Identify the given values
- Mass of the Moon \( M_m = \frac{1}{81} M_e \) (where \( M_e \) is the mass of the Earth)
- Diameter of the Moon \( D_m = \frac{1}{3.7} D_e \) (where \( D_e \) is the diameter of the Earth)

From the diameter, we can find the radius:
- Radius of the Moon \( R_m = \frac{1}{3.7} R_e \)

### Step 2: Write the ratio of gravitational accelerations
Using the formula for gravitational acceleration, we can write the ratio of the gravitational accelerations on the Earth and the Moon:

\[ \frac{g_e}{g_m} = \frac{M_e}{M_m} \cdot \left(\frac{R_m}{R_e}\right)^2 \]

### Step 3: Substitute the known values
Substituting the values we have:
- \( M_m = \frac{1}{81} M_e \) implies \( \frac{M_e}{M_m} = 81 \)
- \( R_m = \frac{1}{3.7} R_e \) implies \( \frac{R_m}{R_e} = \frac{1}{3.7} \)

Thus, we can rewrite the equation:

\[ \frac{g_e}{g_m} = 81 \cdot \left(\frac{1}{3.7}\right)^2 \]

### Step 4: Calculate \( \left(\frac{1}{3.7}\right)^2 \)
Calculating \( \left(\frac{1}{3.7}\right)^2 \):

\[ \left(\frac{1}{3.7}\right)^2 = \frac{1}{13.69} \]

### Step 5: Substitute back into the ratio
Now substituting back into the ratio:

\[ \frac{g_e}{g_m} = 81 \cdot \frac{1}{13.69} \]

### Step 6: Calculate \( g_m \)
We know \( g_e = 9.8 \, \text{m/s}^2 \):

\[ g_m = \frac{g_e}{81 \cdot \frac{1}{13.69}} = \frac{9.8 \cdot 13.69}{81} \]

### Step 7: Perform the calculation
Calculating \( g_m \):

\[ g_m = \frac{9.8 \cdot 13.69}{81} \approx \frac{134.802}{81} \approx 1.66 \, \text{m/s}^2 \]

### Final Answer
Thus, the acceleration due to gravity on the Moon is approximately:

\[ g_m \approx 1.65 \, \text{m/s}^2 \]