Calculate the gravitational potential energy of a body of mass 100 kg at a distance of 6 km from the centre of the earth.

Explanation

To calculate the gravitational potential energy (U) of a body of mass 100 kg at a distance of 6 km from the center of the Earth, we will use the formula for gravitational potential energy:

\[ U = -\frac{G M m}{r} \]

where:
- \( U \) = gravitational potential energy
- \( G \) = universal gravitational constant \( = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \)
- \( M \) = mass of the Earth \( = 6 \times 10^{24} \, \text{kg} \)
- \( m \) = mass of the body \( = 100 \, \text{kg} \)
- \( r \) = distance from the center of the Earth to the body

### Step 1: Convert distance from kilometers to meters
Given that the distance from the center of the Earth is 6 km, we need to convert this to meters:

\[ r = 6 \, \text{km} = 6 \times 1000 \, \text{m} = 6000 \, \text{m} \]

### Step 2: Substitute the values into the formula
Now we can substitute the values into the gravitational potential energy formula:

\[ U = -\frac{(6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2)(6 \times 10^{24} \, \text{kg})(100 \, \text{kg})}{6000 \, \text{m}} \]

### Step 3: Calculate the numerator
First, calculate the numerator:

\[ 6.67 \times 10^{-11} \times 6 \times 10^{24} \times 100 = 6.67 \times 6 \times 100 \times 10^{13} = 4002 \times 10^{13} = 4.002 \times 10^{16} \]

### Step 4: Divide by the distance
Now, divide this result by the distance \( r \):

\[ U = -\frac{4.002 \times 10^{16}}{6000} \]

Calculating this gives:

\[ U = -6.670 \times 10^{12} \, \text{J} \]

### Final Answer
Thus, the gravitational potential energy of the body is:

\[ U \approx -6.67 \times 10^{12} \, \text{J} \]

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