Gravitation

JEE Main level, single correct, medium

51 of 223

Question 51 of 223

Assuming the earth to be a sphere of uniform mass density, a body weighed 300 N300 \mathrm{~N} on the surface of earth. How much it would weigh at R/4 depth under surface of earth ?

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Answer: (D) 225 N225 \mathrm{~N}

 At surface: mg=300 N m=300 gs At Depth R4:gd=gs[1−dR]gd=gs[1−R4R]gd=3 gs4\begin{aligned} & \text { At surface: } \mathrm{mg}=300 \mathrm{~N} \\ & \mathrm{~m}=\frac{300}{\mathrm{~g}_{\mathrm{s}}} \\ & \text { At Depth } \frac{\mathrm{R}}{4}: \mathrm{g}_{\mathrm{d}}=\mathrm{g}_{\mathrm{s}}\left[1-\frac{\mathrm{d}}{\mathrm{R}}\right] \\ & \mathrm{g}_{\mathrm{d}}=\mathrm{g}_{\mathrm{s}}\left[1-\frac{\mathrm{R}}{4 \mathrm{R}}\right] \\ & \mathrm{g}_{\mathrm{d}}=\frac{3 \mathrm{~g}_{\mathrm{s}}}{4}\end{aligned}
weight at depth =m×gd\quad=\mathrm{m} \times \mathrm{g}_{\mathrm{d}}
=m×3 gs4=34×300=225 N\begin{aligned} & =\mathrm{m} \times \frac{3 \mathrm{~g}_{\mathrm{s}}}{4} \\ & =\frac{3}{4} \times 300 \\ & =225 \mathrm{~N}\end{aligned}

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