Physics

Gravitation and Orbital Mechanics

500 Questions

Gravitation and orbital mechanics focus on planetary motion, elliptical orbits, and satellite deployment. Questions examine astrodynamics fundamentals, including geostationary orbits and perturbation theory. These topics are highly relevant for civil services and specialized technical examinations.

Planetary orbitsSatellite dynamicsGeostationary orbitsPerturbation theoryOrbital eccentricity

Gravitation and Orbital Mechanics Questions

Multiple choice evs artificial satellite types of artificial satellites geostationary orbits moon and stars in sky

A communication satellite of earth which takes $24 hr$. to complete one circular orbit eventually has to be replaced by another satellite of double mass. If the new satellites also has an orbital time period of $24 hrs$, then what is the ratio of the radius of the new orbit to the original orbit ?

  1. $1 : 1$
  2. $2 : 1$
  3. $\sqrt 2 : 1$
  4. $1 : 2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Time period of revolution of satellite,  $T= \sqrt{\dfrac{r^3}{GM _e}}$    where $r$ and $M _e$  are the radius  of the orbit and mass of earth.

Thus time period doesn't depend on the mass of satellite.

$\implies $ for same time period, radius of the orbit will be same.
Hence  $r' : r= 1:1$  
Multiple choice evs artificial satellite types of artificial satellites geostationary orbits moon and stars in sky

A body is dropped by a satellite in its geo -stationary orbit.

  1. it will burn on entering in to the atmosphere

  2. it will remain in the same place with respect to the earth
  3. it will reach the earth is $24$ hours

  4. it will perform uncertain motion
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Geo-stationary satellite are those which have same angular velocity as that of earth, so they appear stationary with respect to earth. A body dropped off from satellite would also have same velocity as that of satellite, and so the angular velocity of the body would be same as that of earth also. So it would also seem stationary with respect to earth.

Multiple choice evs artificial satellite types of artificial satellites geostationary orbits moon and stars in sky

The earth satellite can move in an orbit the plane of which coincides with.

  1. The plane of any great circle round the earth

  2. The plane of any latitude circle of the earth

  3. Any plane not containing the centre of the earth

  4. The plane of tropic of cancer
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Such is the case of a geo-stationary satellite, the angular velocity of satellite must be same as that of earth in any circular path around the earth.

Multiple choice evs artificial satellite types of artificial satellites geostationary orbits moon and stars in sky

If $R$ is the average radius of earth, $\omega $ is its angular velocity about its axis and $g$ is the gravitational acceleration on the surface of earth then the cube of the radius of orbit of a geostationary satellite will be equal to.

  1. $\dfrac {R^2g}{\omega }$
  2. $\dfrac {R^2\omega^2 }{g}$
  3. $\dfrac {Rg}{\omega^2 }$
  4. $\dfrac {R^2g}{\omega^2}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
As, $mr\omega^2=\dfrac{GMm}{r^2}$

$\implies r\omega^2=\dfrac{GM}{r^2}$

$\therefore r^3=\dfrac{GM}{\omega^2}=\dfrac{GM}{R^2}\dfrac{R^2}{\omega^2}=g\dfrac{R^2}{\omega^2}$
Multiple choice evs artificial satellite types of artificial satellites geostationary orbits moon and stars in sky

Two planets, $X$ and $Y$, revolving in a orbit around star. Planet $X$ moves in an elliptical orbit whose semi-major axis has length $a$. Planet $Y$ moves in an elliptical orbit whose semi-major axis has a length of $9a$. If planet $X$ orbits with a period $T$, Find out the period of planet $Y$'s orbit?

  1. $729T$
  2. $27T$
  3. $3T$
  4. ${T}/{3}$
  5. ${T}/{27}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Kepler's third law of planetary motion gives    $T^2\propto r^{3}$

    where $T=$ time period of revolution  ,  $r=$ length of semi major axis of elliptical orbit  

by this relation we get  ,  $\dfrac{T _{X}}{T _{Y}}=\dfrac{r _{X}^{3}}{r _{Y}^{3}}$

                                   $\dfrac{T _{X}^2}{T _{Y}^2}=\dfrac{a^{3}}{\left({9a}\right)^{3}}$

but given  $T _{X}=T$

                therefore   $\dfrac{T^2}{T _{Y}^2}=\dfrac{a^{3}}{729a^{3}}$

               or              $T _{Y}=27T$


Multiple choice evs artificial satellite types of artificial satellites geostationary orbits moon and stars in sky

A geosynchronous orbit is one in which the satellite makes one revolution around the Earth in 24 hrs.
How far above the surface of the Earth does this satellite have to orbit?
Assume the radius of the Earth is $6.37 \times {10}^{6} m$, and the mass of the Earth is $5.98 \times {10}^{24} kg$.

  1. $3.59 \times {10}^{7} m$
  2. $4.23 \times {10}^{7} m$
  3. $2.76 \times {10}^{6} m$
  4. $6.37 \times {10}^{6} m$
  5. $1.27 \times {10}^{7} m$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given :   $T = 24$ hrs $ = 84400$  s               $M _e = 5.98 \times 10^{24}$ kg            $R _e  = 0.637 \times 10^7$  m

Time period of satellite moving in orbit of radius $r$            $T = 2\pi \sqrt{\dfrac{r^3}{GM _e}}$

$\therefore$    $86400 = 2\pi \sqrt{\dfrac{r^3}{(6.67 \times 10^{-11}) \times (5.98 \times 10^{24})}}$                         $\implies r  =4.23 \times 10^7$ m

Thus height of satellite above the surface        $h = r - R _e = (4.23 - 0.637) \times 10^7  \approx 3.59 \times 10^7$  m

Multiple choice evs artificial satellite types of artificial satellites geostationary orbits moon and stars in sky

Which of the followings are correct uses of satellite placed in an orbit around Earth?

  1. For observing Earth from a distance

  2. For communication purposes

  3. To forecast weather

  4. All of the above
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Satellite revolves around the earth with fixed a velocity in a circular orbit of certain radius. During its motion, satellite collects informative signals from earth and then sends those signal back to the other part of earth for interpretation. Satellites are used for weather forecasting, communication purposes and observing earth from a distance, GPS etc. 
Multiple choice evs artificial satellite types of artificial satellites geostationary orbits moon and stars in sky

For a satellite to be geostationary, which of the following are essential conditions?

  1. it must always be stationed above the equator

  2. it must be rotate from west to east
  3. it must be about $36,000 km$ above the earth surface

  4. it's orbit must be circular, and not elliptical
Reveal answer Fill a bubble to check yourself
A,B,C,D Correct answer
Explanation

Geostationary satellites revolve in the equatorial plane.
It has to revolve along Earth's rotation direction that is from west to east.
For the time period to be close to $24$ hours, the height of the satellite has to be $36,000\ km$ from earth's surface.
And the orbit is circular.
All options.

Multiple choice evs artificial satellite types of artificial satellites geostationary orbits moon and stars in sky

For geo stationary satelites,

  1. Time period depends on the mass of the satelite

  2. The orbit radius is independent of the mass of earth.

  3. The period is equal to that of the rotation of earth about its axis

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
For geostationary satellites is a circular geosynchronous orbit above earth's equator follows direction of earths rotation ship period is equal to rotation of earth about its axis.
Multiple choice evs artificial satellite types of artificial satellites geostationary orbits moon and stars in sky

A geo-stationary satellite orbits around the earth in a circular orbit of radius $36000\ km$. Then, the time period of a spy satellite orbiting a few $100\ km$ above the earth's surface ($R _{earth}=6400\ km$) will approximately be -

  1. $1/2\ hr$
  2. $1\ hr$
  3. $2\ hr$
  4. $4\ hr$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
Satellite orbits in a radius of $36000㎞\quad \quad { R } _{ 1 }=36000㎞$
${ R } _{ earth }=6400㎞,\quad { R } _{ 2 }=6400$
By Lepler's third law, we know that
${ T }^{ 2 }\propto { R }^{ 3 }$
Also we know that time period of geostationary satellite is $24h$
${ T } _{ 1 }=24h$
$\therefore { \left( \cfrac { { T } _{ 1 } }{ { T } _{ 2 } }  \right)  }^{ 2 }={ \left( \cfrac { { R } _{ 1 } }{ { R } _{ 2 } }  \right)  }^{ 3 }$
$\Rightarrow \cfrac { { \left( 24 \right)  }^{ 2 } }{ { \left( { T } _{ 2 } \right)  }^{ 2 } } ={ \left( \cfrac { 36000 }{ 6400 }  \right)  }^{ 3 }$
${ T } _{ 2 }=24{ \left( \cfrac { 6400 }{ 36000 }  \right)  }^{ 3/2 }$
${ T } _{ 2 }=2h$
Multiple choice evs artificial satellite types of artificial satellites geostationary orbits moon and stars in sky

A geostationary orbit will appear to move in

  1. Equitorial plane

  2. in planes other than equitorial plane

  3. in planes whose angular momentum is not conserved

  4. in planes whose angular momentum is conserved
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A geostationary orbit will appear to move in in planes other than equitorial plane. The time period of such satellites will be 24 hrs only , but they will be at rest only with respect to the equitorial plane and hence such orbits are called parking orbits

The option (b) is the correct option

Multiple choice evs artificial satellite types of artificial satellites geostationary orbits moon and stars in sky

The mean radius of the earth is R, and its angular speed on its axis is $\omega $. What will be the radius of orbit of a geostationary satellite? 

  1. ${\left( {\frac{{Rg}}{{{\omega ^2}}}} \right)^{\frac{1}{3}}}$
  2. ${\left( {\frac{{{R^2}g}}{{{\omega ^2}}}} \right)^{\frac{1}{3}}}$
  3. ${\left( {\frac{{{R^2}g}}{\omega }} \right)^{\frac{1}{3}}}$
  4. ${\left( {\frac{{{R^2}{\omega ^2}}}{g}} \right)^{\frac{1}{3}}}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\dfrac{r^3}{T^2} = \dfrac{Gm}{R^2}. \dfrac{R^2}{4 \pi^2}$


$r^3 = T^2 . (\dfrac{Gm}{R^2}) \dfrac{R^2}{4 \pi^2}$

Since
$g = \dfrac{Gm}{R^2}$

$r^3 = (\dfrac{T^2}{2 \pi} )gR^2$

$r = (\dfrac{R^2g}{w^2})^{1/3}$

Hence (B) is correct answer

Multiple choice evs artificial satellite types of artificial satellites geostationary orbits moon and stars in sky

An instrument package is released from an orbiting earth satellite by simply detaching it from the outer. The package will :

  1. Go away from the earth and get lost in outer space

  2. Fall through a certain distance and then move in an orbit around the earth

  3. Fall towards the surface of earth

  4. Continue moving along with the satellite in the same orbit and with the same velocity
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

When the instrument package is released by simply detaching it from the satellite, then the velocity and acceleration of the package will be same as that of the satellite. So, it continues to moving along with the satellite in the same orbit and with the same velocity. 

Multiple choice evs artificial satellite types of artificial satellites geostationary orbits moon and stars in sky

A geostationary satellite is orbiting the earth at a height of $6R$ above the surface of the earth, where R is the radius of the earth. The time period of another satellite at a height of $2.5R$ from the surface of the earth is $\underline{\hspace{0.5in}}$ hours.

  1. $6.45 h$
  2. $5.39 h$
  3. $6.23 h$
  4. $5.48 h$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

According to Kepler's law,
$\dfrac {T _1^2}{T _2^2}=\dfrac {R _1^3}{R _2^3}$

$\Rightarrow \dfrac {24\times 24}{T _2^2}=\dfrac {6\times 6\times 6\times R^3}{2.5\times 2.5\times 2.5\times R^3}$

$\Rightarrow T _2^2=\dfrac {24\times 24\times 2.5\times 2.5\times 2.5}{6\times 6\times 6}=6.45 h$

Multiple choice evs artificial satellite types of artificial satellites geostationary orbits moon and stars in sky

Statement 1: Geostationary satellites may be setup in equatorial plane in orbits of any radius more than earth's radius.
Statement 2: Geostationary satellites have period of revolution of 24 hrs.

  1. Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1

  2. Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1

  3. Statement-1 is True, Statement-2 is False

  4. Statement-1 is False, Statement-2 is True
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

period of revolution must be 24 hours so radius will be find according to Kepler's law.