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Question

Weight of a body of mass m decreases by $1\% $ when it is raised to height h above the earth’s surface. If the body is taken on a depth $h$ in a mine, change in its weight is

ABDUL JILANI BASHA · Accepted Answer

## CORE INFORMATION
✓ **Answer**: A
## CONCEPT FRAMEWORK
**Primary Concept**: Variation of gravitational acceleration with height and depth.
**Related Topics**: Gravitation, Weight, Percentage Change
**Prerequisites**: Understanding of gravitational force, concept of acceleration due to gravity, approximation using binomial theorem.
## SOLUTION PATHWAY
**Given Information**:
Weight decreases by 1% at height h.
We need to find the percentage change in weight at depth h.
We assume h << R (radius of Earth)
**Method & Steps**:
1.  **Variation of g with height:**
The acceleration due to gravity at height h is given by $g_h = g(1 - \frac{2h}{R})$, where g is the acceleration due to gravity at the surface of the Earth and R is the radius of the Earth. The decrease in weight is 1%, which means the change in g is also 1%. Thus,  $\frac{g-g_h}{g} = 0.01$.  Substituting $g_h$, we get  $\frac{g - g(1-\frac{2h}{R})}{g} = \frac{2h}{R} = 0.01$.

2. **Variation of g with depth:**
The acceleration due to gravity at depth h is given by $g_d = g(1 - \frac{h}{R})$.

3.  **Percentage change at depth:**
We need to find $\frac{g-g_d}{g}$. Substituting $g_d$, we have $\frac{g - g(1 - \frac{h}{R})}{g} = \frac{h}{R}$. From step 1, we found that $\frac{2h}{R} = 0.01$, so $\frac{h}{R} = 0.005$.

4.  **Final Result:**
The percentage change in g at depth h is 0.005, or 0.5%. Since $g_d$ is less than g, the weight decreases. Therefore, the weight decreases by 0.5%.
## OPTION ANALYSIS
[A] ✓: Correct. The weight decreases by 0.5% at depth h, as calculated above.
[B] :x: Incorrect. The weight decrease is not 2%.
[C] :x: Incorrect. The weight does not increase, it decreases.
[D] :x: Incorrect. The weight decreases by 0.5%, not 1%.
## LEARNING AIDS
**Common Mistakes**:
1. **Using incorrect formulas**: Incorrect formulas for variation of g with height and depth can lead to incorrect results. Ensure you are using the correct approximations.
2. **Forgetting the binomial approximation**: The formulas given are approximations valid when h << R. If this condition is not satisfied, these formulas cannot be used.
**Quick Tips**:
1. **Remembering the approximate formulas**: Remember that the change in 'g' at height 'h' is approximately twice that at depth 'h'.
2. **Understanding the relationship between g and weight**: Weight is directly proportional to g, so any change in g will result in a proportional change in weight.

ABDUL JILANI BASHA · Answer

The correct Option is "$0.5\% $ decrease  "

ABDUL JILANI BASHA · Answer

$2\% $ decrease

ABDUL JILANI BASHA · Answer

$0.5\% $ increase

ABDUL JILANI BASHA · Answer

$1\% $ increase

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