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#### Chapters

Chapter 2: Physics and Mathematics

Chapter 3: Rest and Motion: Kinematics

Chapter 4: The Forces

Chapter 5: Newton's Laws of Motion

Chapter 6: Friction

Chapter 7: Circular Motion

Chapter 8: Work and Energy

Chapter 9: Centre of Mass, Linear Momentum, Collision

Chapter 10: Rotational Mechanics

Chapter 11: Gravitation

Chapter 12: Simple Harmonics Motion

Chapter 13: Fluid Mechanics

Chapter 14: Some Mechanical Properties of Matter

Chapter 15: Wave Motion and Waves on a String

Chapter 16: Sound Waves

Chapter 17: Light Waves

Chapter 18: Geometrical Optics

Chapter 19: Optical Instruments

Chapter 20: Dispersion and Spectra

Chapter 21: Speed of Light

Chapter 22: Photometry

## Solutions for Chapter 11: Gravitation

Below listed, you can find solutions for Chapter 11 of CBSE, Karnataka Board PUC HC Verma for Class 11, Class 12 Concepts of Physics Vol. 1.

### HC Verma solutions for Class 11, Class 12 Concepts of Physics Vol. 1 Chapter 11 Gravitation Short Answers [Pages 223 - 224]

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/r^{2} 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?

The weight of a body at the poles is greater than the weight at the equator. Is it the actual weight or the apparent weight we are talking about? Does your answer depend on whether only the earth's rotation is taken into account or the flattening of the earth at the poles is also taken into account?

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.

### HC Verma solutions for Class 11, Class 12 Concepts of Physics Vol. 1 Chapter 11 Gravitation MCQ [Pages 224 - 225]

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

10 m s

^{−2}0⋅0027 m s

^{−2}6⋅4 m s

^{−2}5⋅0 m s

^{−2}.

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

10 m s

^{−2}0⋅0027 m s

^{−2}6⋅4 m s

^{−2}5⋅0 m s

^{−2}

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?

A

B

C

D

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

W

2 W

W/2

2

^{1/3}W at the planet.

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

\[\frac{1}{2}mgR\]

\[2mgR\]

\[mgR\]

\[\frac{1}{4}mgR\]

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

−3 J kg

^{−1}−2 J kg

^{−1}−5 J kg

^{−4}none of these.

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.

Both A and B are correct.

A is correct but B is wrong.

B is correct but A is wrong.

Both A and B are wrong.

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.

Both A and B are correct.

A is correct but B is wrong.

B is correct but A is wrong.

Both A and B are wrong.

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.

Both A and B are correct.

A is correct but B is wrong.

B is correct but A is wrong.

Both A and B are wrong.

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

the mass of the satellite

radius of the orbit

none of them

both of them.

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.

U < K

U > K

U = K

In the Following figure shows the elliptical path of a planet about the sun. The two shaded parts have equal area. If t_{1} and t_{2} be the time taken by the planet to go from a to b and from c to d respectively,

t

_{1}< t_{2}t

_{1}= t_{2}t

_{1}> t_{2}insufficient information to deduce the relation between t

_{1}and t_{2}

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

the earth does not attract the objects in a satellite

the normal force by the chair on the person balances the earth's attraction

the normal force is zero

the person in satellite is not accelerated.

A body is suspended from a spring balance kept in a satellite. The reading of the balance is W_{1} when the satellite goes in an orbit of radius R and is W_{2} when it goes in an orbit of radius 2 −R.

W

_{1}= W_{2}W

_{1}< W_{2}W

_{1}> W_{2}W

_{1 }≠ W_{2}

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

\[\frac{1}{4}\]mgR

\[\frac{1}{2}\]mgR

mgR

2 mgR

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

\[\sqrt{\frac{GM}{4R}}\]

\[\sqrt{\frac{GM}{2R}}\]

\[\sqrt{\frac{GM}{R}}\]

\[\sqrt{\frac{2GM}{R}}\]

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

will be less than \[\nu_e\]

will be more than \[\nu_e\]

will be equal to \[\nu_e\]

will depend on the direction of projection.

### HC Verma solutions for Class 11, Class 12 Concepts of Physics Vol. 1 Chapter 11 Gravitation MCQ [Page 225]

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

increases

decreases

remains constant

oscillates

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.

Speeds of A and B are equal.

The potential energy of earth+A is same as that of earth+B.

The kinetic energy of A and B are equal.

The total energy of earth+A is same as that of earth+B.

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

Speed

Angular speed

Kinetic Energy

Angular momentum.

### HC Verma solutions for Class 11, Class 12 Concepts of Physics Vol. 1 Chapter 11 Gravitation Exercise [Pages 225 - 227]

Two spherical balls of mass 10 kg each are placed 10 cm apart. Find the gravitational force of attraction between them.

Four particles having masses m, 2m, 3m and 4m are placed at the four corners of a square of edge a. Find the gravitational force acting on a particle of mass m placed at the centre.

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.

Three uniform spheres each having a mass M and radius *a* are kept in such a way that each touches the other two. Find the magnitude of the gravitational force on any of the spheres due to the other two.

Four particles of equal masses M move along a circle of radius R under the action of their mutual gravitational attraction. Find the speed of each particle.

Find the acceleration due to gravity of the moon at a point 1000 km above the moon's surface. The mass of the moon is 7.4 × 10^{22} kg and its radius is 1740 km.

Two small bodies of masses 10 kg and 20 kg are kept a distance 1.0 m apart and released. Assuming that only mutual gravitational forces are acting, find the speeds of the particles when the separation decreases to 0.5 m.

A semicircular wire has a length L and mass M. A particle of mass m is placed at the centre of the circle. Find the gravitational attraction on the particle due to the wire.

Derive an expression for the gravitational field due to a uniform rod of length L and mass M at a point on its perpendicular bisector at a distance d from the centre.

Two concentric spherical shells have masses M_{1}, M_{2} and radii R_{1}, R_{2} (R_{1} < R_{2}). What is the force exerted by this system on a particle of mass m_{1} if it is placed at a distance (R_{1}+ R_{2})/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 P_{1} and P_{2 }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.

[**Hint** : If a line y = mx + c makes angle θ with the X-axis, m = tan θ.]

Find the height over the Earth's surface at which the weight of a body becomes half of its value at the surface.

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}.

Find the acceleration due to gravity in a mine of depth 640 m if the value at the surface is 9.800 m s^{−2}. The radius of the earth is 6400 km.

A body is weighed by a spring balance to be 1.000 kg at the North Pole. How much will it weigh at the equator? Account for the earth's rotation only.

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 T_{0}. (a) Find the speed of the ship due to rotation of the earth about its axis. (b) Find the difference between T_{0} 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 time taken by Mars to revolve round the Sun is 1.88 years. Find the ratio of average distance between Mars and the sun to that between the earth and the sun.

The moon takes about 27.3 days to revolve round the earth in a nearly circular orbit of radius 3.84 × 10^{5} km/ Calculate the mass of the earth from these data.

A Mars satellite moving in an orbit of radius 9.4 × 10^{3} km takes 27540 s to complete one revolution. Calculate the mass of Mars.

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 × 10^{24}kg.

(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 × 10^{24} 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 R_{1} and a satellite revolves round it in a circle of radius R_{2}. 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 × 10^{24} 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 × 10^{8} m s^{−1}. What should be the radius of the sphere?

## Solutions for Chapter 11: Gravitation

## HC Verma solutions for Class 11, Class 12 Concepts of Physics Vol. 1 chapter 11 - Gravitation

Shaalaa.com has the CBSE, Karnataka Board PUC Mathematics Class 11, Class 12 Concepts of Physics Vol. 1 CBSE, Karnataka Board PUC solutions in a manner that help students grasp basic concepts better and faster. The detailed, step-by-step solutions will help you understand the concepts better and clarify any confusion. HC Verma solutions for Mathematics Class 11, Class 12 Concepts of Physics Vol. 1 CBSE, Karnataka Board PUC 11 (Gravitation) include all questions with answers and detailed explanations. This will clear students' doubts about questions and improve their application skills while preparing for board exams.

Further, we at Shaalaa.com provide such solutions so students can prepare for written exams. HC Verma textbook solutions can be a core help for self-study and provide excellent self-help guidance for students.

Concepts covered in Class 11, Class 12 Concepts of Physics Vol. 1 chapter 11 Gravitation are The Gravitational Constant, Acceleration Due to Gravity of the Earth, Acceleration Due to Gravity Below and Above the Earth's Surface, Acceleration Due to Gravity and Its Variation with Altitude and Depth, Gravitational Potential Energy, Escape Speed, Earth Satellites, Energy of an Orbiting Satellite, Geostationary and Polar Satellites, Weightlessness, Escape Velocity, Orbital Velocity of a Satellite, Kepler’s Laws, Newton’s Universal Law of Gravitation.

Using HC Verma Class 11, Class 12 Concepts of Physics Vol. 1 solutions Gravitation exercise by students is an easy way to prepare for the exams, as they involve solutions arranged chapter-wise and also page-wise. The questions involved in HC Verma Solutions are essential questions that can be asked in the final exam. Maximum CBSE, Karnataka Board PUC Class 11, Class 12 Concepts of Physics Vol. 1 students prefer HC Verma Textbook Solutions to score more in exams.

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