KKMishra's Tutorials: Solutions to Problems on "GRAVITATION" - H C Verma's Concepts of Physics, Part-I, Chapter-11, OBJECTIVE-II

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GRAVITATION:--
OBJECTIVE - II

1. Let V and E denote the gravitational potential and gravitational field at a point. It is possible to have
(a) V = 0 and E = 0
(b) V = 0 and E ≠ 0
(c) V ≠ 0 and E = 0
(d) V ≠ 0 and E ≠ 0

ANSWER: All

EXPLANATION: Consider a thin spherical shell and choose r = infinity where potential is zero. At infinity V = 0, E = 0. So (a) is true. At any point outside the shell V ≠ 0 and E ≠ 0. So (d) is true. At any point inside the shell E = 0 but V ≠ 0. So (c) is true.
Keeping infinity as a reference point (b) cannot be true. But if we choose reference point just outside the shell then here V = 0 but E ≠ 0. So (b) is true.

2. Inside a uniform spherical shell
(a) the gravitational potential is zero
(b) the gravitational field is zero
(c) the gravitational potential is same everywhere
(d) the gravitational field is same everywhere

ANSWER: (b), (c), (d)

EXPLANATION: Inside a uniform spherical shell the gravitational potential is -GM/a everywhere. Where a = radius of the shell. So (a) is not true but (c) is true. The field is zero and same everywhere inside a shell, so (b) and (d) is true.

3. A uniform spherical shell gradually shrinks maintaining its shape. The gravitational potential at the center
(a) increases
(b) decreases
(c) remains constant
(d) oscillates.

ANSWER: (b)

EXPLANATION: The gravitational potential at the center =-GM/a. When it shrinks 'a' decreases, the numerical value of GM/a increases but this quantity is negative hence increase in numerical value decreases the quantity. Hence (b).

4. Consider a planet moving in an elliptical orbit around the sun. The work done on the planet by the gravitational force of the sun
(a) is zero in any small part of the orbit
(b) is zero in some parts of the orbit
(c) is zero in one complete revolution
(d) is zero in no part of the motion.

ANSWER: (b), (c)

EXPLANATION: When the gravitational force is perpendicular to the direction of motion the work done in that part is zero. In an elliptical orbit, there are points, for examples on the extremes of the major axis where this condition is fulfilled. Hence (b) is true, (d) is not true. At other points, the gravitational force is not perpendicular to the motion, so work done will not be zero, so (a) is not true.

Diagram for Q -4

In one complete revolution, the displacement of the planet is zero, hence the work done is zero, so (c) is true.

5. Two satellites A and B move around the earth in the same orbit. The mass of B is twice the mass of A.
(a) Speeds of A and B are equal
(b) The potential energy of earth+A is same as that of earth +B
(c) The K.E. of A and B are equal
(d) The total energy of earth +A is same as that of earth+B

ANSWER: (a)

EXPLANATION: The velocity of a satellite in an orbit, v = √(GM/a). So the velocity of a satellite in an orbit is in independent of its mass. So (a) is true. The potential energy of the earth satellite system = -GMm/a, so it is dependent on the mass of the satellite. Hence (b) is not true. Since the K.E. (½mv²) depends on the mass, so (c) is not true. Again the total energy of the system = -GMm/2a. It is also dependent on mass, so (d) is not true.

6. Which of the following quantities remain constant in a planetary motion (consider elliptical orbits) as seen from the sun?
(a) Speed
(b) Angular speed
(c) Kinetic energy
(d) Angular momentum.

ANSWER: (d)

EXPLANATION: The speed in an elliptical orbit is not constant and hence not the kinetic energy. So (a) and (c) are not true. The radius vector also does not sweep equal angle in equal time, angular speed is not constant. So (b) is not true. Since there is no external torque on the satellite the angular momentum at any point is constant. (d) is true.

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