Assuming the earth to be a sphere of uniform mass density, the weight of a body at a depth d = R2 from the surface of Earth, if its weight on the surface is 200 N, will be:

Assuming the earth to be a sphere of uniform mass density, the we… — Solution | JEE Physics

MCQEasyJEE 2023Variation with Altitude & Depth

Assuming the earth to be a sphere of uniform mass density, the weight of a body at a depth $d = \frac{R}{2}$ from the surface of Earth, if its weight on the surface is $200 \, \text{N}$ , will be:

A
$500 \, \text{N}$
B
$400 \, \text{N}$
C
$100 \, \text{N}$
D
$300 \, \text{N}$

Answer

Correct answer:C

Step-by-step solution

Standard Method

Given: Weight on the surface is $W_s = 200 \, \text{N}$ . Depth is $d = \frac{R}{2}$ . The Earth is assumed to be a sphere of uniform mass density.

Find: Weight of the body at depth $d$ .

For a body at depth $d$ inside a uniformly dense Earth, weight varies linearly with distance from the center. Hence,

W_d = W_s \times \frac{R-d}{R}

Substitute $W_s = 200 \, \text{N}$ and $d = \frac{R}{2}$ :

W_d = 200 \times \frac{R-\frac{R}{2}}{R}

W_d = 200 \times \frac{\frac{R}{2}}{R}

W_d = 200 \times \frac{1}{2} = 100 \, \text{N}

Therefore, the weight at depth is $100 \, \text{N}$ . The correct option is C. The solution labels option A, but its working gives $100 \, \text{N}$ , which matches option C.

Common mistakes

Using the inverse-square law directly with depth is incorrect here because inside a uniformly dense Earth the enclosed mass changes with radius. Use the linear relation $W_d = W_s \frac{R-d}{R}$ instead.
Taking the depth as distance from the center is wrong. The distance from the center at depth $d$ is $R-d$ , not $d$ . First convert depth below the surface to radius from the center.
Selecting option A from the solution heading without checking the numerical working is a source error. The worked value is $100 \, \text{N}$ , so the matching option is C.

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Answer

Step-by-step solution

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