Two satellites P and Q are moving in different circular orbits around the Earth (radius R). The heights of P and Q from the Earth surface are and , respectively, where . The accelerations of P and Q due to Earth's gravity are and , respectively. If , wh

Question

Two satellites P and Q are moving in different circular orbits around the Earth (radius R). The heights of P and Q from the Earth surface are $h_{P}$ and $h_{Q}$ , respectively, where $h_{P} = \frac{R}{3}$ . The accelerations of P and Q due to Earth's gravity are $g_{P}$ and $g_{Q}$ , respectively. If $\frac{g_{P}}{g_{Q}} = \frac{36}{25}$ , what is the value of $h_{Q}$ ? [2023]

Smart Achievers · Accepted Answer

(1)

Acceleration due to gravity $' g'$ varies with height

$h = \frac{G M}{{(R + h)}^{2}}$

$\frac{g_{p}}{g_{Q}} = \frac{\frac{G M}{{(R + h_{p})}^{2}}}{\frac{G M}{{(R + h_{Q})}^{2}}} \Rightarrow \frac{36}{25} = \frac{{(R + h_{Q})}^{2}}{{(R + h_{p})}^{2}}$

$∴$ $R + h_{Q} = \frac{6}{5} (R + \frac{R}{3})$ $[∵ h_{p} = \frac{R}{3} given]$

$∴$ $h_{Q} = \frac{24}{15} R - R = \frac{9}{15} R = \frac{3}{5} R$

Solution

1 $\frac{3 R}{5}$

2 $\frac{R}{6}$

3 $\frac{6 R}{5}$

4 $\frac{5 R}{6}$

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Solution

1 3R5

2 R6

3 6R5

4 5R6

Ans. (1) Acceleration due to gravity 'g' varies with height h=GM(R+h)2 gpgQ =GM(R+hp)2GM(R+hQ)2⇒3625=(R+hQ)2(R+hp)2 ∴ R+hQ=65(R+R3) [∵hp=R3 given] ∴ hQ=2415R-R=915R=35R

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1 $\frac{3 R}{5}$

2 $\frac{R}{6}$

3 $\frac{6 R}{5}$

4 $\frac{5 R}{6}$