Gravitation Questions and Answers

Example 8 4 Two uniform solid spheres of equal radii R but mass M and 4 M have a centre to centre separation 6 R as shown in Fig 8 10 The two spheres are held fixed A projectile of mass m is projected from the surface of the sphere of mass M directly towards the centre of the second sphere Obtain an expression for the minimum speed v of the projectile so that it reaches the surface of the second sphere M R m 6 R 4M R C

vater The distance of a planet from the sun is 5 times the distance between the earth and the sun The time period of the planet is 1 53 2 years 2 52 3 years 3 51 3 years 151 2 waars

Two uniform sphere each of mass M and radius R separated by a disctance 10R A point mass m projected from surface of one sphere towards other sphere The minimum velocity of projection for which particle will reach at the other sphere 1 3 GM 45R GM 2 8 4 8 GM 45R GM

8 The variations in acceleration due to gravity g of two planets A and B are plotted as a function of distance r from its center Which of the following statements is correct 94 A B 1 The planet A has greater mass than B 2 The planet A and B have same mass MME 3 The planet A is denser than B 4 The planet B is smaller in size

A satellite is seen after each 8 hours over equator at a place on the Earth when its sense of rotation is opposite to the Earth The time interval after which it can be seen at the same place when the sense of rotation of Earth and satellite is same will be O 8 hours O 12 hours O 24 hours 6 hours

Two bodi gravitational potential V at the position where the gravitational field du to them is zero Vis Ov m M Ov Ov GM C

The escape velocity for a planet is ve A particle starts from rest at a large distance from the planet reaches the planet only under gravitational attraction and passes through a smooth tunnel through its centre Its speed at the centre of the planet will be O 1 5v O Ve O Zero

An orbiting satellite will escape if Question Type Single Correct Type 1 Its speed in the orbit is made 1 5 times of its initial values 2 Its kinetic energy in doubled 3 It stops moving in the orbit

1 A binary star consists of two stars A mass 2 2M and B mass 11M where Mis the mass of the sun They are separated by distance d and are rotating about their centre of mass which is stationary The ratio of the total angular momentum of the binary star to the angular momentum of star B about the centre of mass is

The value of g at a particular point on the surface of earth is 9 8 m s2 Suppose the earth suddenly shrinks uniformly to half its present size without losing any mass The value of g at the same point assuming that the distance of the point from the centre of earth does not shrink will now be Question Type Single Correct Type 1 4 9 m sec 2 39 2 m sec 3 9 8 m sec

The figure shows ellipt shaded area SCD is twice the shaded area SAB If t be the planet to move from C to D and to is the time to move from A to O t 8t O t 4t2 O t 2t G D

D A steel wire of diameter 0 8 mm and length 1m is clamped firmly at two points A and B which are 1m apart and in the same plane A body is hung from the middle point of the wire such that the middle point sags 1cm lower from the original position The mass of the body is gm Young s modulus of the material of wire is 2 10 N m

At surface of earth weight of person 72N then his weight at height R 2 from surface of earth is R Radius of earth Question Type Single Correct Type 1 2 3 28N 16N 32N

6 88 A planet moves along an elliptical orbit around the sun At the moment when its distance from the sun is ro its velocity at angle a to ro Find the maximum and minimum distance of this planet from the sun Mass of the sun M Vo is Ans ro 2 n 1 1 4n n 1 sin ro 2 n 1 1 1 4n n 1 sin al

6 92 A meteorite approaching a planet of mass M in the straight line passing through the centre of the planet collides with an automatic space station orbiting the planet in the circular trajectory of radius R The mass of the station is ten times as large as the mass the meteorites As a result of collision the meteorite sticks in the station which goes over to a new orbit with the minimum distance R 2 from the planet Determine the velocity u of the meteorite before the collision

6 83 Masses assumed to be equal to m each hang from strings of different lengths from the ends of a balance on the surface of the earth If the strings have negligible mass and differ in length by h show that the error in weighing W W is given by W 8 W 87 Gmph where p is the density of the earth 3

6 93 Two satellites of the earth move in the same plane with radii a and b b being slightly greater than a What is the minimum interval between the instants when they are on the same line through the centre of the earth i when they move in the same direction ii in opposite direction 5 2 4 3 GM b a Ans 4 5 2 3 GM b a 3 150

6 94 Two lead spheres of 20 cm and 2 cm diameter are placed with their centres 1 0 m apart Calculate the force of attraction between the two spheres The radius of the earth is 6 37 106 m its density is 5 51 x 103 kg m and relative density of lead is 11 5 Ans 1 5 10 10 NJ

6 78 A satellite of mass m is revolving round the earth at a height R above the surface of earth If R is the radius of earth what is the kinetic energy of satellite 1 Ans mgR

A solid spherical planet of mass 2m and radius R has a very small tunnel along its diameter A small cosmic particle of mass m is at a distance 2R from the centre of the planet as shown Both are initially at rest and due to gravitational attraction both start moving toward each other After some time the cosmic particle passes through the centre of the planet Assume the planet and the cosmic particle are isolated from other planets 2m A Displacement of the cosmic particle till that instant is 4R C B Acceleration of the cosmic particle at that instant is zero 3 18Gm C velocity of the cosmic particle at that instant is voo otal work do 2R V 3R 2

6 87 The density of the core of a planet is p and that of the outer shell is p The radii of the core and that of the planet are VR and 2R respectively Gravitational acceleration at the surface of the planet is same as at a depth R Find the P P2 ns P2 3 R P1 P2 Figure 6 114 2R

A particle is projected vertically upwards such th it attains a height of h after 5 sec and 9 sec of motion The speed of projection is g 10 m s 1 20 m sec 2 50 m sec 3 35 m sec 4 70 m sec

E planet A baseball pitcher can throw a fastball at a speed of 150 km hr What is the largest size spherical asteroid of density p 3 g cm 3 from which he can throw the ball fast enough that it a escapes from the asteroid into heliocentric orbit b rises to a height of 50 km

A particle of mass m is placed at the centre of a uniform spherical shell of mass 3 m and radius R The gravitational potential on the surface of the shell is er 1 3 4 Gm R 3Gm R 4Gm R 2Gm R GI 5 150 159 300 9

stars A and B travel in circular orbit around their common centre of mass with radi R and R respectively as shown in figure The time for the star B to move through one orbit is 2 days with it orbital radius R 2 x 10 m The star A has a constant speed of 3 64 x 10 ms in its orbit centre of mass A According to kepler s law Tx r so both have different time periods B the radius of the orbit of the star A R 1 x 10 m C the mass of star A is 3 57 x 10 kg P the mass of star B is 1 79 10 kg

6 89 If gravitational forces between a planet and a satellite is proportional to R 5 2 if R is the orbit radius Then the period of revolution of satellite is proportional to R Find n Ans 7 2

6 41 A planet of mass Mmoves around the Sun along an ellipse so that its minimum distance from the Sun is equal to r and the maximum distance to R Making use of Kepler s laws find its period of revolution frame Ans T r R 2GM

3 A small moon of radius r and mass m is orbiting around a planet of r M in a circular orbit of radius R R r always keeping the same towards the planet If an object on the moon closest to the planet weightlessness find suitable expression for the radius of the orbit Am Y3Y3 with the he

4 A spaceship is launched radially away from the earth with the help of rocket When the spaceship acquires speed to the rocket used switched off As the spaceship proceeds further away from the earth speed v decreases with time t as shown in the graph and eventually spaceship acquires a constant velocity that is one fifth of to How away from the centre of the earth was the rocket switched off Ans 12 toto 5 planet from a

he surface Calculate 6 59 A body is projected horizontally near the surface of the earth with 1 5 times the orbital velocity Calculate the maximum height up to which it will rise above the surface of the earth Ans 2R1

6 79 A spaceship nears the Moon along a parabolic trajectory that almost touches the Moon s surface In order to transfer An into a circular orbit a retro engine fires at the instant of the closest approach The engine ejects gas at a speed of u 4 km s relative to the spaceship in its direction of motion Ifv velocity of spaceship in parabolic trajectory when it almost touches the earth and v velocity of spaceship in circular orbit when it V 2 almost touches the earth Then what fraction of total mass should the fuel burn to transfer space ship to circular orbit 11 12 Ans M Moe T 3600 K 6 C

6 66 The gravitational potential difference between the surface of planet and a point 20 m above the surface is 2 J kg If the gravitational field is uniform then find the work done in carrying a 5 kg body to a height of 4 m above the surface

42 Three particles A B and C of equal mass m ea are placed at the corners of an equilateral triang of side a Force on C is 1 2 m 3 Gm a 2 3Gm a a m MA a a 3

6 86 Four particles each of mass M move along a circle of radius R under the action of their mutual gravitational attractional Find the speed of each particle Ans GM 2 2 1 R 4

Two identical balls are set into motion simultaneously from equal heights h While the ball A is thrown horizontally with velocity v the ball B is just released to fall by itself Choose the alternative that best represents the motion of A and B with respect to an observer who moves with velocity v 2 with respect to the ground as shown in the figure B O O O B St BKKE v 2 h ground

7 A stone is dropped from the top of tall cliff and n seconds later another stone is thrown vertically downwards with a velocity u Then the second stone overtakes the first below the top of the cliff at a distance given by a 8 22 nu 2 u gn 2 u gn 2 gn gn b u 2 2 u gn d g 5 n gn u gn u 1

A satellite is moving in a circular orbit around earth at a height R above earth surface R being radius of earth It s velocity should be increased to k times its initial orbital speed value so as to make it escape from earth gravitational pull and reach infinity Find k Space for rough work

2 Find the height from the surface of the moon where the value of g is equal to the value of g at a height of 57 600 km from the surface of the Earth Take mass of the Earth ME 6 x 1024 kg Mass of the moon M 7 3 1022 kg radius of the Earth RE 6400 and radius of the moon Rm 1740 km 16 to fe

A body of mass 5 kg is suspended by a spring balance on an inclined plane as shown in figure The spring balance reading is g 10 m s 5 kg m 53 50 N 30 N 40 N Smooth

What is the minimum eccentricity of an ellipse that can rest on an inclined plane of angle a with the area vector being perpendicular to the line of greatest slope and the vertical There is uniform gravitational field downwards The coefficient of static friction can be assumed to be practically infinite a

6 A particle of mass m is rotating in a circular orbit of radius r under the action of gravity in the presence of another stationary particle of very large mass M M m Consider that the gravitational potential energy is zero at infinite separation If the total energy of the rotating particle is E then which of the following expressions correctly represents the angular momentum of the particle A r 2Em B T 2Em C rv Em 2 Fm 12

In the figure the block B of mass m starts from rest at the toop of a wedge W of mass M A surfaces are without firction W can slide on the ground B slides down on to the ground moves along it with a speed V has an elastic collision with the wall and climbs back on to W a from the beginnning till the collision with the wall the centre of mass of B plus W does not move horizontally b After the collision the centre of mass of B plus W moves with the velocity 2mV m M c When B reaches its highest position of W the speed of W is d When B reaches its highest position of W the speed of W is 2mV m M mV m M B m W M

A satellite of mass m is in a circular orbit of radius 2R about the earth How much energy is required to transfer it to a circular orbit of radius 4R R Radius of earth Question Type Single Correct Type 1 2 mgR 8 mgR 4

Two satellite A and B having ratio of masses 3 1 are in circular orbits of radius r and 4r Calculate the ratio of total mechanical energies of A to B Question Type Single Correct Type 1 12 2 3

28 A point mass m is placed inside a spherical shell of radius R and mass M at a distance from the centre of 2 R the shell The gravitational force exerted by the shell on the point mass is a GMm R zero b d 2GMm R 4GMm R

A small object mass m falls freely from rest under gravity from a height h and strikes obliquely Fixed smooth inclined plane as shown in figure The impact is elastic r tin A B 1 2u g sin a 3 mnguna 2u sin a g Paragraph Fixed mg 1 The time for which object remains in air between 1st and 2nd impact Fixed inclined plane mysim a 2 ma 2u g 52 The displacement of object along the inclined plane between 1st and 2nd impact will be 1 8h sin a 2 4h sin a 4 4h cosec a 3 8h cosec a 53 The velocity total of object just after the 2nd impact with the inclined plane would be 1 4gh sin a 2 8gh sin a 4 3gh sin a 4 2u cos a g 3 2gh 16gh sin 54 If h h and h3 are the displacements of object along the incline plane between 1st 2nd 2nd 3rc 3rd 4th impact then h h h 1 1 2 3 2 1 3 4

38 Two spheres of masses m and radius a are in contact in space minimum distance of this system from Earth is this minimum distance is called Rochie limit M is mass of the Earth a c a M Earth 1 3 2M m m IR COM 1 3 8M a 2PP Ps a ME Earth 1 1 I 1 3 R r R b d m b 4M m 16M m 39 In the above case two spheres replaced by a satellite of density p can exist near a planet of density p and radius R Rochie limit is 2a bod A dall m 1 3 m a 16PP 1 3 a 2a 1200 c3 1201 m 1 3 R

A geostationary satellite is orbiting around an arbitary planet P at a height of 11R above the surface of P R being the radius of P The time period of another satellite in hours at a height of 2R from the surface of P is P has the time period of 24 hours 1 6 2 2 Official Ans by NTA 3 6 2 3 3 Sol 3 To R3 2 24 12R 3 3 2 T 3hr 4 5

4 A planet has same density and same acceleration due to gravity as of earth and universal gravitational constant G is twice of earth The ratio of their radius a 1 4 b 1 5 c 1 2 d 3 2

The figure shows a small ball with a mass of m attached to one end of an in extensible thread whose other end is connected to a 2 fixed point O at a height of 1 The ball is 3 brought to a horizontal level and released from rest The length of the thread is The horizontal surface is smooth 2 3 1 Magnitude of the impulse on the ball due to the tension as it loses contact with the