Gravitation Questions and Answers

1 Two objects of mass m and 4m are at rest at an infinite separation They move towards each other under mutu gravitational attraction If G is the universal gravitational constant then at separation r A The total energy of the two objects is zero B Their relative velocity of approach is 10Gm C The total kinetic energy of the objects is 1 2 4Gm r in magnitude

1 Four particles A B C and D each of mass m are kept at the corners of a square of side L Now the particle D is taken to infinity by an external agent keeping the other particles fixed at their respective positions The work done by the gravitational force acting on the particle D during its movement is m 1 2 A mo Gm L Gm 2 2 1 m B om DLC Gm L Gm 2 2 1 2 2 4

If potential at the surface of earth is assigned zero value then potential at centre of earth will be Mass M Radius R 1 O 3 3GM 2R 2 4 GM 2R 3GM 2R

Two stars each of one solar mass 2 x 1030 kg are approaching each other for a head on collision When they are 109 km apart their speeds are negligible What is the speed with which they collide The radius of each star is 104 km Assume the stars remain undis torted until they collide Use the known value of G NCERT 2 6 x 106 m s 1

4 At what altitude h above the earth s surface wa the acceleration due to gravity be one fourth o value at the earth s surface 1 h R 2 h 4R 3 h 2R 4 h 16R

A frictionless tunnel is dug inside a uniform sphere as shown in the adjacent figure i A particle of mass m is to be projected from point P so that it can escape from the influence of sphere Calculate the minimum velocity required If the tunnel is dug as shown in the figure ii what will be the effect on the above answer Given R is radius of sphere and M is mass of sphere 45 1 P VN P 45

0 If R and Hare the horizontal range and maximum height attained by a projectile than its speed of projection is 1 2gR 3 2gH 4R gH 8H Rg 2 2gH R g 8H 4 2gH R H 16

1 The density of a newly discovered planet is twice that of earth The acceleration due to gravity at the surface of the planet is equal to that at the surface of the earth If the radius of the earth is R then the radius of the planet would be 1 2R 2 4R 3 1R do 1R 4

5 A body of weight 72 N moves from the surface of earth at a height half of the radius of earth then gravitational force exerted on it will be 1 36 N 2 32 N 3 144 N 4 50 N

A uniform rod of mass M and length is at a 10 The mola distance x from a point particle of mass m The acceleration of particle just after release from the process initial point M ZZZ GMm 1 e x 3 2GM 3 1 x 2 X GM 1 2 x X 2 GM X 1 x 3 m GM 1 4R 3 3R

6 A planet has mass equal to mass of the earth but radius one fourth of radius of the earth Then escape velocity at the surface of this planet will be 1 11 2 km s 2 22 4 km s 3 5 6 km s 4 44 8 km s

Assume that the earth is a uniform sphere of radius R If the acceleration due to gravity at a height h abov the earth s surface is g and at the same depth below the surface of the earth is g h R then show than 82 8 1 1 1

Two bodies of equal masses are some distance apart If 20 of mass is transferred from teh first body to the second body the gravitational force between them 1 Increases by 4 8 2 Increases by 14 3 Decreases by 4 4 Decreases by 14

3 3 5 nm 4 zero A skylab of mass m kg is first launched from the surface of the earth in a circular orbit of radius 2R from the centre of the earth and then it is shifted from this circular orbit to another circular orbit of radius 3R The minimum energy required to place the lab in the first orbit and to shift the lab from first orbit to the second orbit are mgR 31 1 mgR 3 2 mgR mgR 10

2 The figure shows a uniform slab of mass 7 kg What is its moment of inertia in kg cm about the axis normal to the plane of drawing and passing through the point O It is given that the sides OA and OC ar equal i e OA OC

G 1 S G 3 SG S G 19 1 AIPMT Pre 2012 17 A ring is made of a wire having a resistance Ro 12 2 Find the points A and B as shown in the figure at which a current carrying conductor should be connected so that the resistance R of the 2 3 4 8 sub circuit between these points is equal to 32 11 3 l 8 1 12 1 24 5103 N 00 00 2 8 VA 2 R 4 W S S G 1 12 V 2 6 V G S G R 18 If voltage across a bulb rated 220 volts 100 watts drops by 2 5 of its rated value the percentage of the rated value by which the power would decrease is 1 5 2 10 3 20 A 4 2 5 CE0144 In the circuit shown the cells A and B have negligible resistances For VA 12 V R 5000 and R 100 2 the galvanometer G shows no deflection The value of Va is CE0142 B 3 4 V CE0143 4 2V CE0145 ge is to be

A planet of mass m moves in an elliptical or bit around an unknown star of mass M such that its maximum and minimum distances from the star are equal to r and r respec tively The angular momentum of the planet relative to the centre of the star is b 0 a m c m 2GM2 1 1 2GM 1 7112 d 2GMmr 1 12 12

a 18 The earth 82 J cm radiation rate of receives solar min Assuming that the sun radiates like a blackbody calculate the surface temperature of the sun The angle subtended by the sun on the earth is 0 53 and the Stefan constant a 5 67 x 10 W m K Solution D Sun at R learth

34 A satellite of mass m orbit round the earth at a distance of 2 5R above its surface If gravitational field strength at surface of earth is g calculate the centripetal force acting on the satellite 1 3 mg 2 4mg 29 gi GM R2 mv DR 20 13 5 R 2 4 4mg 25 mg 25 sate Ma de max tu g 2 J25 27 1 r 3 57 E GM 8 5Y 8

How much below the surface of earth does acceleration due to gravity become 1 of its value at the earth s sur face The earth is assumed to be a sphere of radius 6 4 x 106 m 6336 km

29 The change in potential energy when a body of mass m is raised to height nR from the earth s surface is R is radius of earth 1 mgR 3 mgR n n 1 n n 1 2 nmgR 4 mgR 2 n n 2 1 Q O

An artificial satellite is moving in a circular orbit around the earth with a speed equal to half the magnitude of escape velocity from the earth i Determine the height of satellite above the earth s surface ii If the satellite is stopped suddenly in its orbit and allowed to fall freely on to the earth find the speed with which it hits the surface of the earth Take g 9 8 m s and 6400 km 7 92 km s 6400 km Re

A man of mass 90 kg is standing in an elevator whose cable broke suddenly If the elevator falls apparent weight of the man is 1 90 N 3 ON 2 90g N 4 any negative value

1 A satellite S is moving in an elliptical orbit around the earth The mass of the satellite is very small compared to the mass of the earth Then 1 the acceleration of S is always directed towards the centre of the earth 2 the angular momentum of S about the centre of the earth changes in direction but its magnitude remains constant 3 the total mechanical energy of S varies periodically with time 4 the linear momentum of S remains constant in

1 If the gravitational potential due to the earth on its surface is assumed to be zero then gravitational potential at height 2R from surface will be Re Radius of the earth g acceleration due to gravity on earth surface 320e 1 3 gRe 3 2gRe 3 2 4 gR 3 2gRe 3 ft bg Re 3 At latitude 3 The escape sp escape speed twice the radi 1 2vo 3 2 18 The radii with de I am 3802 FRIR UR

If the distance between the sun and the earth is 400 times the distance between the moon and the earth and gravitational pull of the sun on the earth is 170 times the gravitational pull of the earth on the moon then the ratio of the mass of the sun to that of the moon is approximately equal to 1 2 7 107 2 4 6 10 3 6 8 10 4 7 4 x 104

33 There are two holes one each along the opposite sides of a wide rectangular tank The cross section of each hole is 0 01 m and the vertical distance between the holes is one metre The tank is filled with water The net force on the tank in newton when the water flows out of the holes is density of water 1000 kgm 1 100 3 300 2 200 4 400 als filled with

A rocket is fired vertically with a speed of 5 km s 1 from the earth s surface How far from the earth does the rocket go before returning to the earth Mass of the earth 6 0 x 1024 kg mean radius of the earth 10 11 N m kg 2 6 4 x 106 m G 6 67 NCERT 8 0 x 106 m from earth s centre

Two heavy spheres each of mass 100 kg and radius 0 10 m are placed 1 0 m apart on a horizontal table What is the gravitational force and potential at the mid point of the line joining the centres of the spheres Is an object placed at that point in equilibrium If so is the equilib rium stable or unstable NCERT 0 2 7 x 10 8 J kg an object placed at the mid point is an unstable equilibrium

lustration 21 A narrow tunnel is dug along the diameter of the earth and a particle of R distance from the centre Find the escape speed mass mo is placed at 2 of the particle from that place R 2

11 A satellite is moving with a constant speed v in circular orbit around the earth An object of mass m is ejected from the satellite such that it just escapes from the gravitational pull of the earth At the time of ejection the kinetic energy of the object is JEE Main 2019 10 Jan Shift 1 3 m b 2 mv c mv d mv a 2

An artificial satellite of mass m of a planet of mass M revolves in a circular orbit whose radius is n times the radius R of planet In the process of motion the satellite experiences a slight resistance of cosmic dust the resistance force being F av where a constant and v is speed Assuming that the orbit remains at most circular Time taken by the satellite to stay in orbit before it falls to the planets surface is n 1 m a n 1 GM R 2 n 1 nx y x y will be

Dis cuss A particle of mass m moves on the axis of a ring of radius R and mass M If particle at rest is released from P then its kinetic energy at centre c will be GMm Smuy M 1 3 GMm R 2GMm R 28 R C3R P 2 m 4 GMm 2R GMm 3R

21 Gravitational potential in a region is given by v 3x 4y 12z J kg The modulus of gravitational field at x 1 y 0 z 3 is 1 20 N kg 2 13 N kg 3 12 N kg 4 5 N kg

A body of mass m is situated at distance 4R abov the earth s surface where R is the radius of eart how much minimum energy be given to the bod so that it may escape Jug in Theme M 1 mgR 3 mgR 5 000 S 2 2mgR 4 mgR 16

A body of mass M is divided into two parts one of mass m and another of mass M m The two pieces are kept at a distance What should be the value of the ratio so that the gravitational force of attraction between the two pieces will be the maximum 1 2 m M

Imagine a light planet revolving around a very heavy star in a circular path The radius of the orbit R and the time period of revolution T If the gravitational force of attraction between the star and the planet is directly proportional to R 5 2 then the relation between T and R would be T2 oc R Determine the value of n 12

Binding energy of a planet and star is increased then 1 Potential energy increased 2 Potential energy decreased 3 Kinetic energy decreased 4 Kinetic and potential energy both remains MO constant

As you have learnt in the text a geostationary satellite orbits the earth at a height of nearly 36 000 km from the surface of the earth What is the potential due to earth s gravity at the site of this satellite Take the potential energy at infinity to be zero Mass of the earth 6 0 x 1024 kg radius 6400 km INCERTI 94 x 106 I kg 1

The mass and the diameter of the planet Jupiter are 309 times and 11 times the mass and the diameter of the earth respectively Determine the acceleration due to gravity on the surface of Jupiter 25 m s 2 S

The radius of the planet Jupiter is 71000 km and the escape velocity from the surface of this planet is 59 6 km s Determine the acceleration due to gravity on the surface of Jupiter 25 m s 2

magnitude of the electric force acting on each charge X 3 An a particle is the nucleus of a helium atom It has a mass m 6 64 10 27 kg and a charg q 2e 3 2 x 10 19 C Compare the force of the electric repulsion between two a particles wit the force of gravitational attraction between them

shows a square current carrying loop ABCD of side 2m and current i A The A magnetic moment M of the loop is Z B A y 30 1 3k A m C i 1 A D paramag 2 k A m

A body of mass m is raised to a height h from a point on the surface of the earth where the acceleration due to gravity is g Prove that the loss of weight due to vari ation of g is 2 mgh where R is the radius of the earth R

As represented in the diagram below an object of mass m located on the surface of the Moon is attracted to the Moon with a gravitational force F Not drawn to scale An object of mass 2m at an altitude equal to the Moon s radius r above the surface of the Moon is attracted to the Moon with a gravitational force of 1 2 F 2F

Two point masses having mass M and 4M are placed at distance r The gravitational potential at a point where gravitational field intensity zero is 1 3 9GM r 3GM 2 4 2GM 3r 6GM 5r

If the kinetic energy of a satellite orbiting around the earth is doubled then A the satellite will escape into the space B the satellite will fall down on the earth C radius of its orbit will be doubled D radius of its orbit will become half

The escape velocity for a planet is v A particle starts from rest at a large distance from the plane the planet only under gravitational attraction and passes through a smooth tunnel through its speed at the centre of the planet will be

A satellite of mass m is orbiting the eart of radius R at a heighth from its surface The tot energy of the satellite in terms of go the value c acceleration due to gravity at the earth s surface is 1 3 2mg R R h mg R 2 R h 2 4 2mg R R h mg R 2 R h

36 The gravitational potential at surface of ea The mass of the earth is M and its radius R Find gravitational potential at centre of earth 1 Zero 2 3 V GM A GM