The mass of the Hubble space telescope is 11600 kg. What is its weight when it is in its orbit 598 km above the earth's surface ? Take R = 6400km.

Explanation

To find the weight of the Hubble Space Telescope when it is in orbit 598 km above the Earth's surface, we can follow these steps:

### Step 1: Understand the formula for weight in orbit
The weight of an object in orbit can be calculated using the formula for gravitational force:
\[ W = m \cdot g' \]
where:
- \( W \) is the weight,
- \( m \) is the mass of the object,
- \( g' \) is the acceleration due to gravity at the height \( h \).

### Step 2: Calculate the acceleration due to gravity at height \( h \)
The formula for the acceleration due to gravity at a height \( h \) above the Earth's surface is given by:
\[ g' = g \cdot \frac{R^2}{(R + h)^2} \]
where:
- \( g \) is the acceleration due to gravity at the Earth's surface (approximately \( 9.81 \, \text{m/s}^2 \)),
- \( R \) is the radius of the Earth (approximately \( 6400 \, \text{km} = 6400 \times 10^3 \, \text{m} \)),
- \( h \) is the height above the Earth's surface (in this case, \( 598 \, \text{km} = 598 \times 10^3 \, \text{m} \)).

### Step 3: Substitute the values into the formula
Now we can substitute the values into the formula:
- \( g = 9.81 \, \text{m/s}^2 \)
- \( R = 6400 \times 10^3 \, \text{m} \)
- \( h = 598 \times 10^3 \, \text{m} \)

Calculating \( R + h \):
\[ R + h = 6400 \times 10^3 + 598 \times 10^3 = 6998 \times 10^3 \, \text{m} \]

Now substituting into the formula for \( g' \):
\[ g' = 9.81 \cdot \frac{(6400 \times 10^3)^2}{(6998 \times 10^3)^2} \]

### Step 4: Calculate \( g' \)
Calculating \( g' \):
1. Calculate \( (6400 \times 10^3)^2 \):
\[ (6400 \times 10^3)^2 = 4.096 \times 10^{13} \]
2. Calculate \( (6998 \times 10^3)^2 \):
\[ (6998 \times 10^3)^2 \approx 4.899 \times 10^{13} \]
3. Now substitute these into the equation for \( g' \):
\[ g' = 9.81 \cdot \frac{4.096 \times 10^{13}}{4.899 \times 10^{13}} \]
\[ g' \approx 9.81 \cdot 0.835 \approx 8.19 \, \text{m/s}^2 \]

### Step 5: Calculate the weight \( W \)
Now we can calculate the weight:
\[ W = m \cdot g' = 11600 \cdot 8.19 \]
\[ W \approx 95184 \, \text{N} \]

### Final Answer
Thus, the weight of the Hubble Space Telescope when it is in orbit 598 km above the Earth's surface is approximately:
\[ W \approx 9.52 \times 10^4 \, \text{N} \]

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