CBSE Class 11 Physics Gravitation Advanced Problems

1. An artificial satellite is describing an equatorial orbit at 1600 km above the surface of the earth, moving in the same direction as the rotation of the earth. Calculate the interval between two successive times at which it will appear vertically overhead to an observer at a fixed point on the equator. Radius of the earth = 6400 km.

2. A satellite is put in an orbit with a speed √1.5 times the speed for a circular orbit at that height. The initial velocity is parallel to the earth’s surface. What will be the maximum distance of the satellite from the centre of the earth if the initial distance is r₁ ?

3. Two planets A and B describe circles of radii r1 and r2 round the sun as centre with speed varying inversely as the square root of their radii. Find the angle between the radii of these two planets when their relative angular velocity is zero.

4. If a planet was suddenly stopped in its orbit supposed to be circular, show that it would fall into the sun in a time √8/ 2 times the period of the planet’s revolution.

5. A satellite is revolving round the earth in a circular orbit of radius r and velocity v₀. A particle is projected from the satellite in forward direction with relative velocity v = √( 5 / 4 −1) v₀. Calculate its mimimum and maximum distances from earth’s centre during subsequent motion of the particle.

6. One of the star of a binary (double) stars system is rotating in a circular orbit of radius r1 with time period T. If the mass of this star is m¹. Find the mass m² of the other star. Also find the distance between the two stars.

Take m1 = 10³⁰ kg, r1 = 2455 × 10⁶ km and T = 20 hrs, G = 6.67 × 10^–11 N-m²/kg²

7. A body is projected vertically upwards from the surface of earth with a velocity sufficient to carry it to infinity. Calculate the time taken by it to reach height h.

8. A space vehicle is in circular orbit about the earth The mass of the vehicle is 300 kg and the radius of the orbit in 2 RE. It is desired to transfer the vehicle to a circular orbit of radius 4RE.
(a) What is the minimum energy required for the transfer?

(b) If the transfer is accomplished through an elliptical orbit as shown in the figure. What initial and final velocity changes are required.
Take g = 10 m/s² at the earth’s surface and RE = 6400 km (radius of the earth).

9. A particle of mass m moves under the action of a central force whose potential is given by V(r) = kr³ (k > 0)
(a) For what energy and angular momentum will the orbit be a circle of radius a about the origin?
(b) What is the period of this circular motion?
(c) If the particle be slightly disturbed from this circular motion, what will be the period of small radial oscillation about r = a?

10. A thin spherical shell of total mass M and radius R is held fixed. There is a small hole in the shell. A mass m is released from rest a distance R from the hole along a line that passes through the hole the centre of the shell. This mass subsequently moves under gravitational force of the shell. How long does the mass take to travelfrom the hole to the point the point diametrically opposite?

11. A spherical hollow is made in a lead sphere of radius R such that its surface touches the outside surface of the lead sphere of radius R/2. The mass of the lead sphere before hollowing was M. What is the force of attraction that remaining mass would exert on a particle of mass m which lies at a distance d from the centre of the lead sphere on the straight line joining the centres of the sphere and the hollow.

12. A point mass m is at a distance x from the centre of a ring of mass M and radius R on its axis as shown in figure. Find the gravitational force between the two. What will this force be if x >> R and x << R. For what value of x is the force maximum.