Gravitation -7

Potential Energy on surface of earth = -
Total energy = mv₀² -
Kinetic Energy at a height h = mv²
Potential Energy at this height = -
Total energy = mv² -
By the principle of conservation of energy

mv² - = mv₀² -
(v₀² - v²) = -
But GM = gR²
(v₀² - v²) =
v₀² - v² =
For maximum height v = 0
v₀ = 90% of escape speed = 0.9


    0.81R + 0.81h_max = h_max

    0.19h_max = 0.81R

    h_max =
    = 4.26R


3. Two satellites A and B of equal masses, move in the equitorial plane of earth, close to the earth's surface. Satellite A moves in the same direction as that of the rotation of earth while satellite B moves in the opposite direction. Determine the ratio of the kinetic energy of B to that of A in the reference frame fixed to earth (g = 9.8 m/s²)
Solution:



Velocity of A with respect to earth = - wR
Velocity of B with respect to earth = + wR
= 1.265
where T = 24 x 60 x 60 s     R = 6400 x 10³ m


HARD:

1. An artificial satellite of mass m of a planet of mass M, resolves in a circular orbit whose radius is n times the radius R of the planet. In the process of motion, the satellite experiences a slight resistance due to cosmic dust. Assuming resistance force on satellite depends on velocity as F = av² where a is constant. Calculate how long the satellite will stay in orbit before it falls onto the planet's surface.
Solution: Total energy in circular orbit of radius r,
E = -
Rate of change of energy = = F_resistiv. v = av².v = av³
also gravity provides the centripetal force

Now



2. Find the maximum and minimum distances of the planet A from the sun S if at a certain moment of time it was a distance r₀ and travelling with the velocity vector being equal to .
Solution:



At minimum and maximum distance velocity of satellite makes on angle of 90⁰ with radius vector.
Applying conservation of angular momentum
mv₀r₀ sin = mvr

By energy conservation

Solving equation (i) and (ii) we get two values of r, one is maximum distance another is minimum distance.
r_max =
r_min =
where


Key words:

• Gravitational Force.
• Gravitationl Field.
• Gravitational Potential.
• Escape Speed.
• Orbittal Speed.
• Areal Velocity.
• Parking Orbit.