Mass of moon is `7.3 xx 10^(22)` kg and its radius is`1.74 xx 10^(6)` m. Find the value of the acceleration due to gravity on the moon.

Options

A. `1.45 N kg^(-1)`
B. `1.55 N kg^(-1)`
C. `1.75 N kg^(-1)`
D. `1.62 N kg^(-1)`

Answer

To find the value of the acceleration due to gravity on the moon, we can use the formula for gravitational acceleration:

\[ g = \frac{G \cdot M}{R^2} \]

Where:
- \( g \) is the acceleration due to gravity,
- \( G \) is the universal gravitational constant, approximately \( 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \),
- \( M \) is the mass of the moon, given as \( 7.3 \times 10^{22} \, \text{kg} \),
- \( R \) is the radius of the moon, given as \( 1.74 \times 10^{6} \, \text{m} \).

### Step 1: Identify the values
- Mass of the moon, \( M = 7.3 \times 10^{22} \, \text{kg} \)
- Radius of the moon, \( R = 1.74 \times 10^{6} \, \text{m} \)
- Gravitational constant, \( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \)

### Step 2: Substitute the values into the formula
Substituting the known values into the formula for \( g \):

\[ g = \frac{(6.67 \times 10^{-11}) \cdot (7.3 \times 10^{22})}{(1.74 \times 10^{6})^2} \]

### Step 3: Calculate the denominator
First, calculate \( R^2 \):

\[ R^2 = (1.74 \times 10^{6})^2 = 1.74^2 \times (10^{6})^2 = 3.0276 \times 10^{12} \, \text{m}^2 \]

### Step 4: Substitute the value of \( R^2 \) back into the equation
Now substitute \( R^2 \) back into the equation for \( g \):

\[ g = \frac{(6.67 \times 10^{-11}) \cdot (7.3 \times 10^{22})}{3.0276 \times 10^{12}} \]

### Step 5: Calculate the numerator
Now calculate the numerator:

\[ 6.67 \times 10^{-11} \cdot 7.3 \times 10^{22} = 4.8671 \times 10^{12} \, \text{N m}^2/\text{kg} \]

### Step 6: Divide the numerator by the denominator
Now divide the numerator by the denominator to find \( g \):

\[ g = \frac{4.8671 \times 10^{12}}{3.0276 \times 10^{12}} \approx 1.607 \, \text{m/s}^2 \]

### Step 7: Final result
Thus, the acceleration due to gravity on the moon is approximately:

\[ g \approx 1.62 \, \text{m/s}^2 \]