What is the value of acceleration due to gravity at a height equal to half the radius of earth, from surface of earth ? [take `g = 10 m//s^(2)` on earth's surface]

Answer

To find the value of acceleration due to gravity at a height equal to half the radius of the Earth, we can use the formula for gravitational acceleration at a height \( h \) above the Earth's surface:

\[ g' = \frac{g}{(1 + \frac{h}{r})^2} \]

Where:
- \( g' \) is the acceleration due to gravity at height \( h \),
- \( g \) is the acceleration due to gravity at the Earth's surface (given as \( 10 \, \text{m/s}^2 \)),
- \( h \) is the height above the Earth's surface,
- \( r \) is the radius of the Earth.

### Step 1: Identify the height \( h \)
Given that the height \( h \) is equal to half the radius of the Earth, we can express this as:
\[ h = \frac{r}{2} \]

### Step 2: Substitute \( h \) into the formula
Now, we substitute \( h \) into the formula for \( g' \):
\[ g' = \frac{g}{(1 + \frac{h}{r})^2} = \frac{g}{(1 + \frac{r/2}{r})^2} \]

### Step 3: Simplify the expression
Simplifying the term \( \frac{h}{r} \):
\[ \frac{h}{r} = \frac{r/2}{r} = \frac{1}{2} \]
So, we can rewrite the formula:
\[ g' = \frac{g}{(1 + \frac{1}{2})^2} = \frac{g}{(1.5)^2} \]

### Step 4: Calculate \( (1.5)^2 \)
Calculating \( (1.5)^2 \):
\[ (1.5)^2 = 2.25 \]

### Step 5: Substitute \( g \) and calculate \( g' \)
Now, substituting \( g = 10 \, \text{m/s}^2 \):
\[ g' = \frac{10}{2.25} = \frac{10 \times 4}{9} = \frac{40}{9} \]

### Step 6: Calculate the final value
Calculating \( \frac{40}{9} \):
\[ g' \approx 4.44 \, \text{m/s}^2 \]

### Final Answer
Thus, the acceleration due to gravity at a height equal to half the radius of the Earth is approximately:
\[ g' \approx 4.44 \, \text{m/s}^2 \]

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