Test Index

Gravitation Part 1

Section: Physics

Question : 5 of 100

Marks: +1, -0

R_{E}

r < R_{E}

Solution:

Given that, radius of Earth

=R_{E}

Height above Earth’s surface, h = r

Depth below Earth surface, d = r

We know that, acceleration due to gravity at height (r) and depth (r) is (for

r < R_{E}

)

g_{h}=g\left(\frac{R_{E}}{R_{E}+h}\right)^{2}=g\left(\frac{R_{E}}{R_{E}+r}\right)^{2}

...(i)

and

g_{d}=g\left(\frac{R_{E}-d}{R_{E}}\right)=g\left(\frac{R_{E}-r}{R_{E}}\right)

...(ii)

\therefore \frac{g_{d}}{g_{h}}=\frac{g\left(\frac{R_{E}-r}{R_{E}}\right)}{g\left(\frac{R_{E}}{R_{E}+r}\right)^{2}}

[from Eqs. (i) and (ii)]

\Rightarrow \frac{g_{d}}{g_{h}}=\frac{(R_{E}-r)(R_{E}+r)^{2}}{R^{3}}=\frac{R_{E}^{3}+R_{E}^{2}r-R_{E}r^{2}-r^{3}}{R_{E}^{3}}

1+\frac{r}{R_{E}}-\frac{r^{2}}{R_{E}^{2}}-\frac{r^{3}}{R_{E}^{3}}

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