At a height h above the earth's surface the fractional change in the value of acceleration due to gravity is :
[Given R - Radius of earth]

Question

Accepted Answer

To determine the fractional change in the value of acceleration due to gravity at a height $ h $ above the Earth's surface, we start by considering the formula for gravitational acceleration at a distance $ r $ from the center of the Earth:

$$ g' = \frac{GM}{(R+h)^2} $$

where $ G $ is the gravitational constant, $ M $ is the mass of the Earth, and $ R $ is the radius of the Earth. At the Earth's surface, the acceleration due to gravity is:

$$ g = \frac{GM}{R^2} $$

The fractional change in gravity, $\Delta g/g$, when moving from the surface to a height $ h $ is given by:

$$ \frac{\Delta g}{g} = \frac{g - g'}{g} = 1 - \frac{g'}{g} $$

Substituting the expressions for $ g $ and $ g' $:

$$ \frac{\Delta g}{g} = 1 - \frac{R^2}{(R+h)^2} $$

For small heights $ h $ compared to the Earth's radius $ R $, we can use the binomial approximation:

$$ (R+h)^2 \approx R^2 + 2Rh $$

Thus:

$$ \frac{R^2}{(R+h)^2} \approx \frac{R^2}{R^2 + 2Rh} \approx 1 - \frac{2h}{R} $$

Therefore, the fractional change in gravity is approximately:

$$ \frac{\Delta g}{g} \approx \frac{2h}{R} $$

This matches Option 3: $\frac{2h}{R}$.

**Explanation for Other Options:**

1. **Option 1: $\frac{h}{R}$** - This option suggests a linear relationship between the fractional change in gravity and height, which is incorrect. The correct relationship involves a factor of 2, as derived above.

2. **Option 2: $1-\frac{2h}{R}$** - This option incorrectly suggests that the fractional change is a subtraction from 1, which does not align with the derived formula. The correct expression is a positive fractional change, not a reduction from 1.

3. **Option 4: $1+\frac{2h}{R}$** - This option implies an increase in gravity with height, which contradicts the physical reality that gravity decreases with increasing distance from the Earth's center.

In conclusion, Option 3 is the correct answer because it accurately represents the fractional change in gravity at a height $ h $ above the Earth's surface, as derived from the principles of gravitational physics.