HC Verma solutions for Concepts of Physics Vol. 1 [English] Class 11 and 12 chapter 11 - Gravitation [Latest edition]

Can two particles be in equilibrium under the action of their mutual gravitational force? Can three particles be? Can one of the three particles be?

Is there any meaning of "Weight of the earth"?

If heavier bodies are attracted more strongly by the earth, why don't they fall faster than the lighter bodies?

Can you think of two particles which do not exert gravitational force on each other?

The earth revolves round the sun because the sun attracts the earth. The sun also attracts the moon and this force is about twice as large as the attraction of the earth on the moon. Why does the moon not revolve round the sun? Or does it?

At noon, the sun and the earth pull the objects on the earth's surface in opposite directions. At midnight, the sun and the earth pull these objects in same direction. Is the weight of an object, as measured by a spring balance on the earth's surface, more at midnight as compared to its weight at noon?

An apple falls from a tree. An insect in the apple finds that the earth is falling towards it with an acceleration g. Who exerts the force needed to accelerate the earth with this acceleration g?

Suppose the gravitational potential due to a small system is k/r² at a distance r from it. What will be the gravitational field? Can you think of any such system? What happens if there were negative masses?

The gravitational potential energy of a two-particle system is derived in this chapter as `"U"=("Gm"_1"m"_2)/"r"`. Does it follow from this equation that the potential energy for \[r = \infty\] must be zero? Can we choose the potential energy for \[r = \infty\] to be 20 J and still use this formula? If no, what formula should be used to calculate the gravitational potential energy at separation r?

The weight of an object is more at the poles than at the equator. Is it beneficial to purchase goods at equator and sell them at the pole? Does it matter whether a spring balance is used or an equal-beam balance is used?

If the radius of the earth decreases by 1% without changing its mass, will the acceleration due to gravity at the surface of the earth increase or decrease? If so, by what per cent?

A nut becomes loose and gets detached from a satellite revolving around the earth. Will it land on the earth? If yes, where will it land? If no, how can an astronaut make it land on the earth?

Is it necessary for the plane of the orbit of a satellite to pass through the centre of the earth?

Consider earth satellites in circular orbits. A geostationary satellite must be at a height of about 36000 km from the earth's surface. Will any satellite moving at this height be a geostationary satellite? Will any satellite moving at this height have a time period of 24 hours?

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?

Two satellites going in equatorial plane have almost same radii. As seen from the earth one moves from east one to west and the other from west to east. Will they have the same time period as seen from the earth? If not which one will have less time period?

A spacecraft consumes more fuel in going from the earth to the moon than it takes for a return trip. Comment on this statement.

The acceleration of moon with respect to earth is 0⋅0027 m s⁻² and the acceleration of an apple falling on earth' surface is about 10 m s⁻². Assume that the radius of the moon is one fourth of the earth's radius. If the moon is stopped for an instant and then released, it will fall towards the earth. The initial acceleration of the moon towards the earth will be

The acceleration of the moon just before it strikes the earth in the previous question is

Suppose, the acceleration due to gravity at the earth's surface is 10 m s⁻² and at the surface of Mars it is 4⋅0 m s⁻². A 60 kg passenger goes from the earth to the Mars in a spaceship moving with a constant velocity. Neglect all other objects in the sky. Which part of the following figure best represents the weight (net gravitational force) of the passenger as a function of time?

Consider a planet in some solar system which has a mass double the mass of the earth and density equal to the average density of the earth. An object weighing W on the earth will weight

If the acceleration due to gravity at the surface of the earth is g, the work done in slowly lifting a body of mass m from the earth's surface to a height R equal to the radius of the earth is

A person brings a mass of 1 kg from infinity to a point A. Initially the mass was at rest but it moves at a speed of 2 m s ⁻¹ as it reaches A. The work done by the person on the mass is −3 J. The potential at A is

Let V and E be the gravitational potential and gravitational field at a distance r from the centre of a uniform spherical shell. Consider the following two statements :

(A) The plot of V against r is discontinuous.
(B) The plot of E against r is discontinuous.

Let V and E represent the gravitational potential and field at a distance r from the centre of a uniform solid sphere. Consider the two statements:
(A) the plot of V against r is discontinuous.
(B) The plot of E against r is discontinuous.

Take the effect of bulging of earth and its rotation in account. Consider the following statements :

(A) There are points outside the earth where the value of g is equal to its value at the equator.
(B) There are points outside the earth where the value of g is equal to its value at the poles.

The time period of an earth-satellite in circular orbit is independent of

The magnitude of gravitational potential energy of the moon-earth system is U with zero potential energy at infinite separation. The kinetic energy of the moon with respect to the earth is K.

In the Following figure shows the elliptical path of a planet about the sun. The two shaded parts have equal area. If t₁ and t₂ be the time taken by the planet to go from a to b and from c to d respectively,

A person sitting in a chair in a satellite feels weightless because

A body is suspended from a spring balance kept in a satellite. The reading of the balance is W₁ when the satellite goes in an orbit of radius R and is W₂ when it goes in an orbit of radius 2 −R.

The kinetic energy needed to project a body of mass m from the earth' surface to infinity is

A particle is kept at rest at a distance R (earth's radius) above the earth's surface. The minimum speed with which it should be projected so that it does not return is 

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 v_e. Its speed with respect to the satellite

Let V and E denote the gravitational potential and gravitational field at a point. It is possible to have

(a) \[V = 0 \text { and }E = 0\]

(b) \[V = 0 \text { and } E \neq 0\]

(c) \[V \neq 0 \text { and }E = 0\]

(d) \[V \neq 0 \text { and }E \neq 0\]

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

A uniform spherical shell gradually shrinks maintaining its shape. The gravitational potential at the centre

Consider a planet moving in an elliptical orbit round 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.

Two satellites A and B move round the earth in the same orbit. The mass of B is twice the mass of A.

Which of the following quantities remain constant in a planetary motion (consider elliptical orbits) as seen from the sun?

Three equal masses m are placed at the three corners of an equilateral triangle of side a. Find the force exerted by this system on another particle of mass m placed at (a) the mid-point of a side, (b) at the centre of the triangle.

Two concentric spherical shells have masses M₁, M₂ and radii R₁, R₂ (R₁ < R₂). What is the force exerted by this system on a particle of mass m₁ if it is placed at a distance (R₁+ R₂)/2 from the centre?

A tunnel is dug along a diameter of the earth. Find the force on a particle of mass m placed in the tunnel at a distance x from the centre.

A tunnel  is dug along a chord of the earth at a perpendicular distance R/2 from the earth's centre. The wall of the tunnel may be assumed to be frictionless. Find the force exerted by the wall on a particle of mass m when it is at a distance x from the centre of the tunnel.

A solid sphere of mass m and radius r is placed inside a hollow thin spherical shell of mass M and radius R as shown in the following figure . A particle of mass m' is placed on the line joining the two centres at a distance x from the point of contact of the sphere and the shell. Find the magnitude of the resultant gravitational force on this particle due to the sphere and the shell if (a) r < x < 2r, (b) 2r < x < 2R and (c) x > 2R.

A uniform metal sphere of radius a and mass M is surrounded by a thin uniform spherical shell of equal mass and radius 4a (In the following figure). The centre of the shell falls on the surface of the inner sphere. Find the gravitational field at the points P₁ and P₂ shown in the figure.

A thin spherical shell having uniform density is cut in two parts by a plane and kept separated as shown in the following figure. The point A is the centre of the plane section of the first part and B is the centre of the plane section of the second part. Show that the gravitational field at A due to the first part is equal in magnitude to the gravitational field at B due to the second part.

Two small bodies of masses 2.00 kg and 4.00 kg are kept at rest at a separation of 2.0 m. Where should a particle of mass 0.10 kg be placed to experience no net gravitational force from these bodies? The particle is placed at this point. What is the gravitational potential energy of the system of three particles with usual reference level?

Three particles of mass m each are placed at the three corners of an equilateral triangle of side a. Find the work which should be done on this system to increase the sides of the triangle to 2a.

A particle of mass 100 g is kept on the surface of a uniform sphere of mass 10 kg and radius 10 cm. Find the work to be done against the gravitational force between them to take the particle away from the sphere.

The gravitational field in a region is given by

\[\vec{E} = \left( 5 N {kg}^{- 1} \right) \vec{i} + \left( 12 N {kg}^{- 1} \right) \vec{j}\] . (a) Find the magnitude of the gravitational force acting on a particle of mass 2 kg placed at the origin. (b) Find the potential at the points (12 m, 0) and (0, 5 m) if the potential at the origin is taken to be zero. (c) Find the change in gravitational potential energy if a particle of mass 2 kg is taken from the origin to the point (12 m, 5 m). (d) Find the change in potential energy if the particle is taken from (12 m, 0) to (0, 5 m).

The gravitational potential in a region is given by V = 20 N kg⁻¹ (x + y). (a) Show that the equation is dimensionally correct. (b) Find the gravitational field at the point (x, y). Leave your answer in terms of the unit vectors \[\vec{i} , \vec{j} , \vec{k}\] . (c) Calculate the magnitude of the gravitational force on a particle of mass 500 g placed at the origin.

The gravitational field in a region is given by \[E = \left( 2 \overrightarrow{i} + 3 \overrightarrow{j} \right) N {kg}^{- 1}\] . Show that no work is done by the gravitational field when a particle is moved on the line 3y + 2x = 5.

What is the acceleration due to gravity on the top of Mount Everest? Mount Everest is the highest mountain peak of the world at the height of 8848 m. The value at sea level is 9.80 m s⁻².

A body stretches a spring by a particular length at the earth's surface at the equator. At what height above the south pole will it stretch the same spring by the same length? Assume the earth to be spherical.

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.

The moon takes about 27.3 days to revolve round the earth in a nearly circular orbit of radius 3.84 × 10⁵ km/ Calculate the mass of the earth from these data.

A satellite of mass 1000 kg is supposed to orbit the earth at a height of 2000 km above the earth's surface. Find (a) its speed in the orbit, (b) is kinetic energy, (c) the potential energy of the earth-satellite system and (d) its time period. Mass of the earth = 6 × 10²⁴kg.

(a) Find the radius of the circular orbit of a satellite moving with an angular speed equal to the angular speed of earth's rotation. (b) If the satellite is directly above the North Pole at some instant, find the time it takes to come over the equatorial plane. Mass of the earth = 6 × 10²⁴ kg.

What is the true weight of an object in a geostationary satellite that weighed exactly 10.0 N at the north pole?

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.

Find the minimum colatitude which can directly receive a signal from a geostationary satellite.

A particle is fired vertically upward from earth's surface and it goes up to a maximum height of 6400 km. Find the initial speed of particle.

A particle is fired vertically upward with a speed of 15 km s⁻¹. With what speed will it move in interstellar space. Assume only earth's gravitational field.

A mass of 6 × 10²⁴ kg (equal to the mass of the earth) is to be compressed in a sphere in such a way that the escape velocity from its surface is 3 × 10⁸ m s⁻¹. What should be the radius of the sphere?