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Question

If the radius of the earth were to shrink by $1\% $ its mass remaining same, the acceleration due to gravity on the earth’s surface would

P.NARASIMHA · Accepted Answer

## CORE INFORMATION
✓ **Answer**: C
## CONCEPT FRAMEWORK
	**Primary Concept**: Acceleration due to gravity
	**Related Topics**: Newton's Law of Universal Gravitation, Percentage Change
	**Prerequisites**: Understanding of the formula for acceleration due to gravity.
## SOLUTION PATHWAY
**Given Information**:
	* Radius of Earth shrinks by 1%
	* Mass of Earth remains constant
**Method & Steps**:
	1. **Initial Acceleration due to gravity:**
		Reasoning: The acceleration due to gravity (g) on the surface of the Earth is given by the formula: $g = \frac{GM}{R^2}$, 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. **New Radius:**
		Working: If the radius shrinks by 1%, the new radius (R') will be R' = R - 0.01R = 0.99R
		Key Point:  The mass remains constant, so M' = M
	3. **New Acceleration due to gravity:**
		Working: The new acceleration due to gravity (g') will be: $g' = \frac{GM}{(0.99R)^2} = \frac{GM}{0.9801R^2} = \frac{1}{0.9801} \cdot \frac{GM}{R^2} = \frac{1}{0.9801}g$
	4. **Percentage Change in g:**
		Working: $g' \approx 1.0203 g$. The percentage change is $\frac{g'-g}{g} * 100 =  \frac{1.0203g - g}{g} * 100 = 0.0203 * 100 = 2.03%$.  This is approximately 2% increase.
		Result: The acceleration due to gravity increases by approximately 2%.
		Verification: The change in gravity is inversely proportional to the square of the radius. A small decrease in radius leads to a larger increase in g. Since radius decreased by 1%, acceleration due to gravity increases by approximately 2%.
 ## OPTION ANALYSIS
[A] :x: Decrease by 2% is incorrect. The acceleration due to gravity increases due to a decrease in radius.
[B] :x: Remain unchanged is incorrect. The radius has changed, so the acceleration due to gravity will change.
[C] ✓: Increase by 2% is correct. The calculated value is approximately a 2% increase.
[D] :x: Become zero is incorrect. The acceleration due to gravity will still be non-zero as long as mass exists.
 ## LEARNING AIDS
	**Common Mistakes**:
		1. **Using linear change**: Assuming the change in g is directly proportional to change in R. The relationship is inverse square, g is proportional to 1/R^2
		2. **Incorrectly calculating percentage change**: Forgetting to multiply by 100 or not considering the initial value when computing % change.
	**Quick Tips**:
		1.  Use the approximation: If a quantity x changes by a small percentage p, then x^2 changes by approximately 2p, x^3 changes by approximately 3p, and 1/x changes by approximately -p and so on.
		2. Remember the formula for the acceleration due to gravity: $g = \frac{GM}{R^2}$

P.NARASIMHA · Answer

Decrease by $2\% $

P.NARASIMHA · Answer

Remain unchanged

P.NARASIMHA · Answer

The correct Option is "Increase by $2\% $ "

P.NARASIMHA · Answer

Become zero

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