Correct formula for height of a satellite from Earth's surface is

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Q.
Correct formula for height of a satellite from Earth's surface is _____. [2024]

1 ${(\frac{T^{2} R^{2} g}{4 π^{2}})}^{1 / 3} - R$

2 ${(\frac{T^{2} R^{2} g}{4 π^{2}})}^{1 / 2} - R$

3 ${(\frac{T^{2} R^{2}}{4 π^{2} g})}^{1 / 3} - R$

4 ${(\frac{T^{2} R^{2} g}{4 π^{2}})}^{- 1 / 3} + R$

Ans.
(1)

$\frac{m v^{2}}{(R + h)} = \frac{G M m}{{(R + h)}^{2}} \Rightarrow v = \sqrt{\frac{G M}{R + h}}$

$Now T = \frac{2 π (R + h)}{\sqrt{\frac{G M}{R + h}}}$

$\Rightarrow T = \frac{2 π {(R + h)}^{3 / 2}}{\sqrt{G M}} = \frac{2 π {(R + h)}^{3 / 2}}{R \sqrt{g}}$

$\Rightarrow T^{2} R^{2} g = 4 π^{2} (R + h^{3})$

$\Rightarrow {(\frac{T^{2} R^{2} g}{4 π^{2}})}^{1 / 3} - R = h$

Solution

Q.
Correct formula for height of a satellite from Earth's surface is _____. [2024]

1 ${(\frac{T^{2} R^{2} g}{4 π^{2}})}^{1 / 3} - R$

2 ${(\frac{T^{2} R^{2} g}{4 π^{2}})}^{1 / 2} - R$

3 ${(\frac{T^{2} R^{2}}{4 π^{2} g})}^{1 / 3} - R$

4 ${(\frac{T^{2} R^{2} g}{4 π^{2}})}^{- 1 / 3} + R$

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Solution

Q. Correct formula for height of a satellite from Earth's surface is _____. [2024]

1 (T2R2g4π2)1/3-R

2 (T2R2g4π2)1/2-R

3 (T2R24π2g)1/3-R

4 (T2R2g4π2)-1/3+R

Ans. (1) mv2(R+h)=GMm(R+h)2⇒v=GMR+h Now T=2π(R+h)GMR+h ⇒T=2π(R+h)3/2GM=2π(R+h)3/2Rg ⇒T2R2g=4π2(R+h3) ⇒(T2R2g4π2)1/3-R=h

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Q.
Correct formula for height of a satellite from Earth's surface is _____. [2024]

1 ${(\frac{T^{2} R^{2} g}{4 π^{2}})}^{1 / 3} - R$

2 ${(\frac{T^{2} R^{2} g}{4 π^{2}})}^{1 / 2} - R$

3 ${(\frac{T^{2} R^{2}}{4 π^{2} g})}^{1 / 3} - R$

4 ${(\frac{T^{2} R^{2} g}{4 π^{2}})}^{- 1 / 3} + R$