4. If the value of g at the surface of the Earth is \( 9.8 \, \text{m/s}^2 \), what is its approximate value at a height equal to the radius of the Earth?
To find the value of \( g \) at a height equal to the Earth's radius (\( h = R \)), we use the formula:
\( g_h = g \left( \frac{R}{R + h} \right)^2 \)
Substituting \( h = R \):
\( g_h = g \left( \frac{R}{R + R} \right)^2 \)
Simplify:
\( g_h = g \left( \frac{R}{2R} \right)^2 \)
\( g_h = g \left( \frac{1}{2} \right)^2 \)
\( g_h = g \cdot \frac{1}{4} \)
Substituting \( g = 9.8 \, \text{m/s}^2 \):
\( g_h = 9.8 \cdot \frac{1}{4} = 2.45 \, \text{m/s}^2 \)
Thus, the approximate value of \( g \) at a height equal to the Earth's radius is:
\( g_h = 2.45 \, \text{m/s}^2 \)
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