Find the acceleration due to gravity of the moon at a point 1000 km above the moon's surface. The mass of the moon is `7.4xx10^2` kg and its radius is 1740 km.

Explanation

To find the acceleration due to gravity on the moon at a point 1000 km above its surface, we can follow these steps:

### Step 1: Understand the formula for gravitational acceleration
The acceleration due to gravity \( g \) at a distance \( r \) from the center of a mass \( M \) is given by the formula:
\[ g = \frac{G \cdot M}{r^2} \]
where:
- \( G \) is the gravitational constant, approximately \( 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \)
- \( M \) is the mass of the moon
- \( r \) is the distance from the center of the moon to the point where we want to find \( g \)

### Step 2: Calculate the total distance from the center of the moon
The radius of the moon is given as 1740 km, and we need to add the height above the surface (1000 km) to this radius:
\[ r = \text{radius of the moon} + \text{height above the surface} = 1740 \, \text{km} + 1000 \, \text{km} = 2740 \, \text{km} \]
Convert this distance into meters:
\[ r = 2740 \, \text{km} \times 1000 \, \text{m/km} = 2.74 \times 10^6 \, \text{m} \]

### Step 3: Substitute the values into the formula
Now we can substitute the values into the gravitational acceleration formula:
\[ g = \frac{G \cdot M}{r^2} \]
Substituting \( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \), \( M = 7.4 \times 10^{22} \, \text{kg} \), and \( r = 2.74 \times 10^6 \, \text{m} \):
\[ g = \frac{6.67 \times 10^{-11} \cdot 7.4 \times 10^{22}}{(2.74 \times 10^6)^2} \]

### Step 4: Calculate \( r^2 \)
Calculate \( r^2 \):
\[ r^2 = (2.74 \times 10^6)^2 = 7.5076 \times 10^{12} \, \text{m}^2 \]

### Step 5: Calculate the numerator
Now calculate the numerator:
\[ 6.67 \times 10^{-11} \cdot 7.4 \times 10^{22} = 4.9388 \times 10^{12} \, \text{N m}^2/\text{kg} \]

### Step 6: Calculate \( g \)
Now substitute the values into the equation for \( g \):
\[ g = \frac{4.9388 \times 10^{12}}{7.5076 \times 10^{12}} \approx 0.657 \, \text{m/s}^2 \]

### Final Answer
Thus, the acceleration due to gravity on the moon at a point 1000 km above its surface is approximately:
\[ g \approx 0.657 \, \text{m/s}^2 \]

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