Mass of moon is `7.349 xx 10^(22) kg` and its radius is `1.738 xx 10^(6) m`. Calculate its mean density and acceleration due to gravity on its surface. Given `G = 6668 xx 10^(-11) Nm^(2) kg ^(-2)`.

Explanation

To solve the problem, we need to calculate the mean density of the moon and the acceleration due to gravity on its surface using the given values.

### Step 1: Calculate the Volume of the Moon
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Where:
- \( r \) is the radius of the moon.

Given:
- Radius of the moon \( r = 1.738 \times 10^6 \, \text{m} \)

Substituting the radius into the volume formula:
\[ V = \frac{4}{3} \pi (1.738 \times 10^6)^3 \]

### Step 2: Calculate the Mean Density
Density \( \rho \) is defined as mass per unit volume:
\[ \rho = \frac{M}{V} \]
Where:
- \( M \) is the mass of the moon.

Given:
- Mass of the moon \( M = 7.349 \times 10^{22} \, \text{kg} \)

Substituting the values:
\[ \rho = \frac{7.349 \times 10^{22}}{V} \]

### Step 3: Calculate the Acceleration due to Gravity
The formula for acceleration due to gravity \( g \) at the surface of a celestial body is given by:
\[ g = \frac{GM}{R^2} \]
Where:
- \( G \) is the universal gravitational constant.
- \( R \) is the radius of the moon.

Given:
- \( G = 6.668 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2 \)
- Radius of the moon \( R = 1.738 \times 10^6 \, \text{m} \)

Substituting the values into the formula:
\[ g = \frac{6.668 \times 10^{-11} \times 7.349 \times 10^{22}}{(1.738 \times 10^6)^2} \]

### Step 4: Calculate the Final Values
Now, we can compute the values from the equations derived above.

1. **Volume Calculation**:
\[ V = \frac{4}{3} \pi (1.738 \times 10^6)^3 \approx 4.034 \times 10^{18} \, \text{m}^3 \]

2. **Density Calculation**:
\[ \rho = \frac{7.349 \times 10^{22}}{4.034 \times 10^{18}} \approx 1825.67 \, \text{kg/m}^3 \]

3. **Acceleration due to Gravity Calculation**:
\[ g = \frac{6.668 \times 10^{-11} \times 7.349 \times 10^{22}}{(1.738 \times 10^6)^2} \approx 1.622 \, \text{m/s}^2 \]

### Final Answers
- Mean Density of the Moon: \( \approx 1825.67 \, \text{kg/m}^3 \)
- Acceleration due to Gravity on the Moon's Surface: \( \approx 1.622 \, \text{m/s}^2 \)