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Question

The ratio of acceleration due to gravity at a height 3R above earth’s surface to the acceleration due to gravity on the surface of the earth is (R = radius of earth)

Mr. ANIL KUMAR.G · Accepted Answer

## CORE INFORMATION
✓ **Answer**: C
## CONCEPT FRAMEWORK
	**Primary Concept**: Newton's Law of Gravitation and its application to acceleration due to gravity.
	**Related Topics**: Gravitational force, inverse square law.
	**Prerequisites**: Understanding of Newton's law of gravitation and the concept of acceleration due to gravity.
## SOLUTION PATHWAY
**Given Information**:
	* Height above Earth's surface: h = 3R
	* Radius of Earth: R

**Method & Steps**:
	1. **Acceleration due to gravity on the surface of the Earth:**
		Reasoning: The acceleration due to gravity on the surface of the earth is given by g = GM/R², where G is the gravitational constant, M is the mass of the Earth, and R is the radius of the Earth.
		Formula/Rule: $g = \frac{GM}{R^2}$
	2. **Acceleration due to gravity at a height h above the surface:**
		Reasoning: The acceleration due to gravity at a height h above the surface is given by $g_h = \frac{GM}{(R+h)^2}$. In this case, h=3R, so the distance from the center of the earth is (R+3R) = 4R.
		Formula/Rule: $g_h = \frac{GM}{(R+h)^2}$
	3. **Calculate $g_h$ at h=3R:**
		Working: $g_{3R} = \frac{GM}{(R+3R)^2} = \frac{GM}{(4R)^2} = \frac{GM}{16R^2}$
		Key Point: The distance from the center of the Earth is crucial.
	4. **Find the ratio of $g_{3R}$ to g:**
		Working:  Ratio = $\frac{g_{3R}}{g} = \frac{GM/16R^2}{GM/R^2} = \frac{1}{16}$
		Result: The ratio is 1/16
		Verification: The calculation confirms the correct answer.
 ## OPTION ANALYSIS
[A] :x: Incorrect. This would be the ratio if the height was R and the distance was 2R. 
[B] :x: Incorrect. This would be the ratio if the height was R and the distance was 2R, and then not squared. 
[C] ✓: Correct. The ratio of the acceleration due to gravity at a height 3R to that on the surface is 1/16.
[D] :x: Incorrect. This is not the correct ratio.
 ## LEARNING AIDS
	**Common Mistakes**:
		1. **Using incorrect distance:** Using just 3R as the distance from the center of the Earth instead of 4R.  Always remember to add the Earth's radius to the height above the surface.
		2. **Incorrectly squaring the distance:** Forgetting to square the distance in the formula.
	**Quick Tips**:
		1. Remember that acceleration due to gravity decreases with the square of the distance from the center of the planet.
		2. Always express the distance from the center of the planet to correctly calculate the acceleration due to gravity at different altitudes.

Mr. ANIL KUMAR.G · Answer

$\frac{1}{9}$

Mr. ANIL KUMAR.G · Answer

$\frac{1}{4}$

Mr. ANIL KUMAR.G · Answer

The correct Option is "$\frac{1}{{16}}$"

Mr. ANIL KUMAR.G · Answer

$\frac{1}{3}$

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