Test: Earth satellites & Energy of an orbiting satellite (8 August) - JEE MCQ

Concept:

• The time period of the satellite: It is the time taken by the satellite to complete one revolution around the Earth.
• Consider a satellite orbiting the earth at a height h from the surface of the earth of radius R.
• The circumference of orbit of satellite = 2π(R+h)
• The orbital velocity of the satellite at a height h is given by:

Where M is the mass of the Earth.

• The kinetic energy of the satellite of mass m in a circular orbit is:

• The potential energy at distance (R +h) from the centre of the earth is given by:

• The total energy of a circularly orbiting satellite ​is

Explanation:

• A negative sign in the value of energy shows that the nature of force between satellite and Earth is attractive.
• Thus, the total energy of a circularly orbiting satellite is negative.

• The total energy of the earth satellite bounded system is negative = (−GM/ 2R)
  where R is the radius of the satellite and M is the mass of the earth.
• The negative sign indicates that the force of attraction between satellite and earth is continuously reduced due to atmospheric friction (viscous force).
• As a result, the radius of the orbit or height decreases gradually, and ultimately it comes back to earth with increasing speed and burns in the atmosphere.

Hence, option (3) is the correct answer.

Satellite:

• Satellites are the natural or artificial bodies that describe an orbit around a planet by its gravitational force of attraction.
• Moon is a natural satellite.

Kinetic energy

• The energy possessed by a body due to the virtue of its motion is called kinetic energy.

Gravitational potential energy:

• The work done in bringing anybody from infinity to a point in the gravitational field is called the gravitational potential energy of that body at that point.
• The gravitational potential energy at a height h from the earth's surface is given as,

Where U = gravitational potential energy, KE = kinetic energy, M = mass of the earth, m = mass of the body, R = radius of the earth, v = velocity, and h = height from the surface of the earth

Calculations:

• We know that the kinetic energy of the satellite is given as,

• We know that the potential energy of the satellite is given as,

Where r = distance of the satellite from the center of the planet

By equation 1 and equation 2,

Hence, option 1 is correct.

Concept:

• The time period of the satellite: It is the time taken by the satellite to complete one revolution around the Earth.
• Consider a satellite orbiting the earth at a height h from the surface of the earth of radius R.
• The circumference of orbit of satellite = 2π(R+h)
• The orbital velocity of the satellite at a height h is given by:

Where M is the mass of the Earth.

• The kinetic energy of the satellite of mass m in a circular orbit is:

• The potential energy at distance (R +h) from the centre of the earth is given by:

• The total energy of a circularly orbiting satellite ​is

Explanation:

The relationship between the total energy (E) of an orbiting satellite and its kinetic energy (KE) is given by:

E = -KE.

Orbital velocity of the satellite is given by

• From the above equation, it is clear that the orbital velocity of the satellite is independent of the mass of the satellite. 
• Speed of the ball will be the same as that of the satellite, so when the ball is dropped from a satellite revolving around the earth at height of 120 Km, it will continue to move with the same speed along the original orbit of the satellite. Therefore option 2 is correct.

Angular momentum is the rotational equivalent of linear momentum.
Angular momentum (L) for a point object accelerating with respect to a fixed point at a distance R is given by:
L = R x mv

Where m is the mass of the body, and v is the velocity.

So, for a satellite rotating about the Earth, angular momentum ∝ R
The potential energy of an orbiting satellite is inversely proportional to the radial distance from the centre of the earth i.e. (R+h).
So, potential energy ∝ R^-1

​Therefore, none of the given options is correct.