The correct formula for the height h of a sat | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) For a satellite of mass $m$ orbiting at a height $h$ above the Earth's surface,the gravitational force provides the necessary centripetal force:$\

Vedclass · Accepted Answer

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

Vedclass · Answer

(A) For a satellite of mass $m$ orbiting at a height $h$ above the Earth's surface,the gravitational force provides the necessary centripetal force:$\frac{GMm}{(R+h)^2} = \frac{mv^2}{(R+h)}$$\Rightarrow \frac{GM}{(R+h)} = v^2 \quad \dots(1)$Since $v = (R+h)\omega = (R+h)\frac{2\pi}{T}$,we have $v^2 = (R+h)^2 \frac{4\pi^2}{T^2} \quad \dots(2)$Also,the acceleration due to gravity at the Earth's surface is $g = \frac{GM}{R^2}$,so $GM = gR^2 \quad \dots(3)$Substituting $(2)$ and $(3)$ into $(1)$:$\frac{gR^2}{(R+h)} = (R+h)^2 \frac{4\pi^2}{T^2}$$\Rightarrow (R+h)^3 = \frac{gR^2 T^2}{4\pi^2}$$\Rightarrow R+h = \left(\frac{gR^2 T^2}{4\pi^2}\right)^{1/3}$$\Rightarrow h = \left(\frac{T^2 R^2 g}{4\pi^2}\right)^{1/3} - R$

The correct formula for the height $h$ of a satellite from the Earth's surface is:

Similar Questions

Two satellites are revolving around the Earth with orbital velocities $v_1$ and $v_2$ at radii $r_1$ and $r_2$ respectively,where $r_1 > r_2$. Then:

Two satellites $A$ and $B$ are orbiting in circular paths of radii $4R$ and $R$ respectively. If the orbital velocity of satellite $A$ is $3V$,what is the orbital velocity of satellite $B$ (in $V$)?

Does the orbital velocity of a planet depend on its mass?

$A$ geostationary satellite is taken to a new orbit such that its distance from the centre of the earth is doubled. Find the time period of this satellite in the new orbit.

The orbit of a geo-stationary satellite is circular. The time period of the satellite depends on:
$(i)$ mass of the satellite
(ii) mass of the earth
(iii) radius of the orbit
(iv) height of the satellite from the surface of the earth
Which of the following is correct?

Vedclass Test Series

Exam Paper Generator

Online Exam Module