Advertisements
Advertisements
प्रश्न
Answer the following question in detail.
Obtain an expression for the binding energy of a satellite revolving around the Earth at a certain altitude.
Advertisements
उत्तर
An expression for the binding energy of satellite revolving in a circular orbit around the Earth:
- Consider a satellite of mass m revolving at height h above the surface of the Earth in a circular orbit. It possesses potential energy as well as kinetic energy.
- Let M be the mass of the Earth, R be the Radius of the Earth, vc be the critical velocity of satellite, r = (R + h) be the radius of the orbit.
- Kinetic energy of satellite = `1/2 "mv"_"c"^2 = 1/2 "GMm"/"r"`
- The gravitational potential at a distance r from the centre of the Earth is `- "GM"/"r"`
∴ The potential energy of a satellite
= Gravitational potential × mass of a satellite
= `- "GMm"/"r"` - The total energy of satellite is given as
T.E. = K.E. + P.E.
`= 1/2 "GMm"/"r" - "GMm"/"r" = - 1/2 "GMm"/"r"` - The total energy of a circularly orbiting satellite is negative. The negative sign indicates that the satellite is bound to the Earth, due to the gravitational force of attraction. For the satellite to be free from the Earth’s gravitational influence its total energy should become zero or positive.
- Hence the minimum energy to be supplied to unbind the satellite is `+ 1/2 "GMm"/"r"`. This is the binding energy of a satellite.
APPEARS IN
संबंधित प्रश्न
Suppose there existed a planet that went around the sun twice as fast as the earth.What would be its orbital size as compared to that of the earth?
No part of India is situated on the equator. Is it possible to have a geostationary satellite which always remains over New Delhi?
As the earth rotates about its axis, a person living in his house at the equator goes in a circular orbit of radius equal to the radius of the earth. Why does he/she not feel weightless as a satellite passenger does?
A satellite is orbiting the earth close to its surface. A particle is to be projected from the satellite to just escape from the earth. The escape speed from the earth is ve. Its speed with respect to the satellite
At what rate should the earth rotate so that the apparent g at the equator becomes zero? What will be the length of the day in this situation?
A pendulum having a bob of mass m is hanging in a ship sailing along the equator from east to west. When the ship is stationary with respect to water the tension in the string is T0. (a) Find the speed of the ship due to rotation of the earth about its axis. (b) Find the difference between T0 and the earth's attraction on the bob. (c) If the ship sails at speed v, what is the tension in the string? Angular speed of earth's rotation is ω and radius of the earth is R.
A Mars satellite moving in an orbit of radius 9.4 × 103 km takes 27540 s to complete one revolution. Calculate the mass of Mars.
The radius of a planet is R1 and a satellite revolves round it in a circle of radius R2. The time period of revolution is T. Find the acceleration due to the gravitation of the planet at its surface.
State the conditions for various possible orbits of satellite depending upon the horizontal/tangential speed of projection.
Derive an expression for the critical velocity of a satellite.
Draw a labelled diagram to show different trajectories of a satellite depending upon the tangential projection speed.
Answer the following question in detail.
What is a critical velocity?
Describe how an artificial satellite using a two-stage rocket is launched in an orbit around the Earth.
A planet has mass 6.4 × 1024 kg and radius 3.4 × 106 m. Calculate the energy required to remove an object of mass 800 kg from the surface of the planet to infinity.
The ratio of energy required to raise a satellite of mass 'm' to a height 'h' above the earth's surface of that required to put it into the orbit at same height is ______.
[R = radius of the earth]
A geostationary satellite is orbiting the earth at the height of 6R above the surface of earth. R being radius of earth. The time period of another satellite at a height of 2.5 R from the surface of earth is ____________.
Assuming that the earth is revolving around the sun in circular orbit of radius 'R', the angular momentum is directly proportional to rn. The value of 'n' is ______.
If a body weighing 40 kg-wt is taken inside the earth to a depth to `1/2` th radius of the earth, then the weight of the body at that point is ____________.
Out of following, the only correct statement about satellites is ____________.
A satellite of mass 'm' is revolving around the earth of mass 'M' in an orbit of radius 'r' with constant angular velocity 'ω'. The angular momentum of the satellite is ______.
(G =gravitational constant)
In the case of earth, mean radius is 'R', acceleration due to gravity on the surface is 'g', angular speed about its own axis is 'ω'. What will be the radius of the orbit of a geostationary satellite?
A satellite is to revolve round the earth in a circle of radius 9600 km. The speed with which this satellite be projected into an orbit, will be ______.
A geostationary satellite is orbiting the earth at a height 6R above the surface of the earth, where R is the radius of the earth. This time period of another satellite at a height (2.5 R) from the surface of the earth is ______.
The period of revolution of a satellite is ______.
An artificial satellite is moving in a circular orbit around the earth with a speed equal to half the magnitude of escape velocity from the earth. If the satellite is stopped in its orbit and allowed to fall freely onto the earth, the speed with which it hits the surface ______ km/s.
[g = 9.8 ms-2 and Re = 6400 km]
A satellite revolves around a planet very close to its surface. By what maximum factor can its kinetic energy be increased suddenly, such that it revolves in orbit in the same way?
A satellite is revolving around a planet in a circular orbit close to its surface and ρ is the mean density and R is the radius of the planet, then the period of ______.
(G = universal constant of gravitation)
Two satellites of same mass are orbiting round the earth at heights of r1 and r2 from the centre of earth. Their potential energies are in the ratio of ______.
What is the approximate period of revolution for the Moon, Earth's only natural satellite?
Artificial satellites are launched for all the following purposes EXCEPT:
