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JEE Main 202131 AugustEvening Shift
Exam
JEE Main
Year
2021
Date
31 August
Shift
Evening Shift
Subject
Physics
Chapter
Gravitation

JEE Main 2021 · 31 August · Evening Shift · Physics · Gravitation

If R _ E be the radius of Earth, then the ratio between the...

If RE{R}_{E} be the radius of Earth, then the ratio between the acceleration due to gravity at a depth rr below and a height rr above the earth surface is:
(Given : r<REr < {R}_{E})If R _ E be the radius of Earth, then the ratio between the acceleration due to gravity at a depth r below and a height

Options

  1. A1+rRE−r2RE2−r3RE31 + \dfrac{r}{{R}_{E}} - {r}^{2}{R}_{E}^{2} - {r}^{3}{R}_{E}^{3}
  2. B1+rRE+r2RE2+r3RE31 + \dfrac{r}{{R}_{E}} + {r}^{2}{R}_{E}^{2} + {r}^{3}{R}_{E}^{3}
  3. C1−rRE−r2RE2−r3RE31 - \dfrac{r}{{R}_{E}} - {r}^{2}{R}_{E}^{2} - {r}^{3}{R}_{E}^{3}
  4. D1+rRE−r2RE2+r3RE31 + \dfrac{r}{{R}_{E}} - {r}^{2}{R}_{E}^{2} + {r}^{3}{R}_{E}^{3}

Correct answer

A. 1+rRE−r2RE2−r3RE31 + \dfrac{r}{{R}_{E}} - {r}^{2}{R}_{E}^{2} - {r}^{3}{R}_{E}^{3}

Step-by-step solution

If R _ E be the radius of Earth, then the ratio between the acceleration due to gravity at a depth r below and a height
ghgd=GM(RE+r)2GMRE3(RE−r)=RE3(RE+r)2(RE−r)\dfrac{{g}_{h}}{{g}_{d}} = \dfrac{\dfrac{GM}{{\left({R}_{E} + r\right)}^{2}}}{GM{R}_{E}^{3}\left({R}_{E} - r\right)} = {R}_{E}^{3}{\left({R}_{E} + r\right)}^{2}\left({R}_{E} - r\right)
gdgh=(RE+r)\dfrac{{g}_{d}}{{g}_{h}} = {\left({R}_{E} + r\right)}^{2}(RE−r)RE3=RE3−REr2+rRE2−r3RE3=1−r2RE2+rRE−(rRE)3\left({R}_{E} - r\right){R}_{E}^{3} = {R}_{E}^{3} - {R}_{E}{r}^{2} + r{R}_{E}^{2} - {r}^{3}{R}_{E}^{3} = 1 - {r}^{2}{R}_{E}^{2} + \dfrac{r}{{R}_{E}} - {\left(\dfrac{r}{{R}_{E}}\right)}^{3}
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