Question

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

• A. $0.5\%$ decrease.
• B. $2\%$ decrease.
• C. $0.5\%$ increase.
• D. $1\%$ increase.

Answer 1

The correct option is A

$0.5\%$ decrease.

Step 1: Given data

The weight of a body of mass m decreases by $1\%$ when it is raised to a height $h$ above the earth's surface

Step 2: To find

The change in weight if the body is taken to a depth h in a mine.

Step 3: Formula used

Acceleration due to gravity at a height h,

$g\text{'}=g\left(1-\frac{2h}{R}\right)$

Acceleration due to gravity at a depth h,

$g\text{'}\text{'}=g\left(1-\frac{h}{R}\right)$

where $R$ is radius of earth.

Step 4: Calculation

Acceleration due to gravity when the body is raised to a height $h$,

$g\text{'}=g\left(1-\frac{2h}{R}\right)$

Given that the weight of the body decreases by $1\%$ in this case.

Therefore, weight of the body at a height $h$,

$$\begin{gathered} w\text{'}=mg\text{'} \\ w\text{'}=mg\left(1-\frac{2h}{R}\right) \end{gathered}$$

Original weight of the body,

$w=mg$

Since it is given that the weight decreases by $1\%$,

$$\begin{gathered} w\text{'}=\frac{99}{100}\times w \\ \Rightarrow mg\left(1-\frac{2h}{R}\right)=\frac{99}{100}\times mg \\ \Rightarrow \left(1-\frac{2h}{R}\right)=\frac{99}{100} \\ \Rightarrow \frac{2h}{R}=1-\frac{99}{100} \\ \Rightarrow \frac{2h}{R}=\frac{1}{100} \\ \Rightarrow h=\frac{R}{200} \end{gathered}$$

When the body is taken to a depth $h$,

Acceleration due to gravity,

$g\text{'}\text{'}=g\left(1-\frac{h}{R}\right)$

weight in this case,

$$\begin{gathered} w\text{'}\text{'}=mg\text{'}\text{'} \\ w\text{'}\text{'}=mg\left(1-\frac{h}{R}\right) \\ w\text{'}\text{'}=mg\left(1-\frac{R}{200R}\right)\left(\because h=\frac{R}{200}\right) \\ w\text{'}\text{'}=mg\left(1-\frac{1}{200}\right) \\ w\text{'}\text{'}=mg\frac{199}{200} \\ w\text{'}\text{'}=\frac{199}{200}w \\ w\text{'}\text{'}=\frac{99.5}{100}w \end{gathered}$$

Here the weight at depth $h$ is $99.5\%$ of the initial weight.

Therefore, the weight decreases by $0.5\%$.

Option A is correct.