Calculate the percentage change in weight of a body if taken to a height of 10 km above the surface of earth. The radius of earth is 6,400 km.

Answer

To calculate the percentage change in weight of a body when taken to a height of 10 km above the surface of the Earth, we can follow these steps:

### Step 1: Write down the given data
- Height (h) = 10 km = 10,000 m (since we need to convert km to m for calculations)
- Radius of the Earth (R) = 6,400 km = 6,400,000 m
- Acceleration due to gravity at the surface of the Earth (g) = 9.8 m/s²

### Step 2: Use the formula for acceleration due to gravity at height
The formula to calculate the acceleration due to gravity (g') at a height (h) above the Earth's surface is given by:
\[ g' = g \left(1 - \frac{2h}{R}\right) \]

### Step 3: Substitute the values into the formula
Substituting the values we have:
\[ g' = 9.8 \left(1 - \frac{2 \times 10,000}{6,400,000}\right) \]

### Step 4: Calculate the fraction
First, calculate the fraction:
\[ \frac{2 \times 10,000}{6,400,000} = \frac{20,000}{6,400,000} = 0.003125 \]

### Step 5: Calculate g'
Now substitute this back into the equation for g':
\[ g' = 9.8 \left(1 - 0.003125\right) \]
\[ g' = 9.8 \times 0.996875 \]
\[ g' \approx 9.769375 \, \text{m/s}^2 \]

### Step 6: Calculate the change in weight
The change in weight is proportional to the change in acceleration due to gravity. The percentage change in weight can be calculated using:
\[ \text{Percentage Change} = \frac{g - g'}{g} \times 100\% \]

### Step 7: Substitute the values
Substituting the values we have:
\[ \text{Percentage Change} = \frac{9.8 - 9.769375}{9.8} \times 100\% \]

### Step 8: Calculate the percentage change
Calculate the difference:
\[ 9.8 - 9.769375 \approx 0.030625 \]

Now, calculate the percentage:
\[ \text{Percentage Change} = \frac{0.030625}{9.8} \times 100\% \approx 0.3125\% \]

### Final Answer
The percentage change in weight of the body when taken to a height of 10 km above the surface of the Earth is approximately **0.316%**.

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