Gravitation PYQs Set 9 (PDF)

Set 9 of 12 · Questions 241-270 of 351

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Mere Teacher · Previous Year Questions JEE Main Physics · Gravitation · Set 9 (Q241-270)
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241. $An object is kept at rest at a distance of 3 R above the earth's surface where R is earth's radius. The minimum speed with which it must be projected so that it does not return to earth is: (Assume M = mass of earth, G = Universal gravitational constant)$ (1) $\frac{3 GM}{R}$ (2) $\frac{2 GM}{R}$ (3) $\frac{GM}{2 R}$ (4) $\frac{GM}{R}$ 242. $A body is projected vertically upwards from the surface of earth with a velocity sufficient enough to carry it to infinity. The time taken by it to reach height h is S.$ (1) $\frac{R _{e}}{2 g} ((1 + \frac{h}{R _{e}})^{3/2} - 1)$ (2) $\frac{2 R _{e}}{g} ((1 + \frac{h}{R _{e}})^{3/2} - 1)$ (3) $\frac{1}{3} \frac{R _{e}}{2 g} ((1 + \frac{h}{R _{e}})^{3/2} - 1)$ (4) $\frac{1}{3} \frac{2 R _{e}}{g} ((1 + \frac{h}{R _{e}})^{3/2} - 1)$ 243. $The escape velocity of a body on the surface of the earth is 11.2 k m / s . If the earth’s mass increases to twice its present value and radius of the earth becomes half, the escape velocity becomes$ (1) $22.4 k m / s$ (2) $44.8 k m / s$ (3) $5.6 k m / s$ (4) $11.2 k m / s .$ 244. $If the gravitational field in the space is given as (- \frac{K}{r ^{2}}) . Taking the reference point to be at r = 2 cm with gravitational potential V = 10 J/kg. Find the gravitational potential at r = 3 cm in SI unit (Given, that K = 6 J cm/kg)$ (1) $9$ (2) $11$ (3) $12$ (4) $10$ 245. $A particle of mass M is situated at the center of spherical shell of same mass M and radius a. The magnitude of the gravitational potential at a point situated at a/2 distance from the center, will be:$ (1) $\frac{2 GM}{a}$ (2) $\frac{3 GM}{a}$ (3) $\frac{4 GM}{a}$ (4) $\frac{GM}{a}$ 246. $The acceleration due to gravity g and mean density of the earth p are related by which of the following relations? (where G is the gravitational constant and R is the radius of the earth.)$ (1) $p = \frac{3 g}{4 π GR}$ (2) $p = \frac{3 g}{4 π G R ^{3}}$ (3) $p = \frac{4 π g R ^{2}}{3 G}$ (4) $p = \frac{4 π g R ^{3}}{3 G} .$ 247. $Two satellites A and B of masses 200 kg and 400 kg are revolving round the earth at height of 600 km and 1600 km respectively. If T_{A} and T_{B} are the time periods of A and B respectively then the value of T_{B} - T_{A} : [Given: radius of earth = 6400 km, mass of earth (= 6 \times 10^{24} kg)$ (1) $4.24 \times 10^{2} s$ (2) $3.33 \times 10^{2} s$ (3) $1.33 \times 10^{3} s$ (4) $4.24 \times 10^{3} s$ 248. $Starting from the center of the earth having radius r, the variation of g (acceleration due to gravity) is shown by$ (1) (2) (3) (4) 249. $The orbital angular momentum of a satellite is L, when it is revolving in a circular orbit at height h from earth surface. If the distance of satellite from the earth centre is increased by eight times to its initial value, then the new angular momentum will be$ (1) $8L$ (2) $9L$ (3) $4L$ (4) $3L$ 250. $The angular momentum of a planet of mass M moving around the sun in an elliptical orbit is L . Themagnitude of the areal velocity of the planet is:$ (1) $\frac{4 L}{M}$ (2) $\frac{L}{M}$ (3) $\frac{2 L}{M}$ (4) $\frac{L}{2 M}$ 251. $A particle of mass M is situated at the centre of a spherical shell of same mass and radius a . The magnitude of the gravitational potential at a point situated at a l 2 distance from the center, will be$ (1) $\frac{GM}{a}$ (2) $\frac{2 GM}{a}$ (3) $\frac{3 GM}{a}$ (4) $\frac{4 GM}{a}$ 252. $Two bodies of mass m and 9 m are placed at a distance R . The gravitational potential on the line joining the bodies where the gravitational field equals zero, will be (G = gravitational constant) [NEET 2023]$ (1) $- \frac{12 Gm}{R}$ (2) $- \frac{16 Gm}{R}$ (3) $- \frac{20 Gm}{R}$ (4) $- \frac{8 Gm}{R}$ 253. $A planet of mass m moves in a elliptical orbit around an unknown star of mass M such that its maximum and minimum distances from the star are equal to r_{1} and r_{2} respectively. The angular momentum of the planet relative to the centre of the star is$ (1) $m \frac{2 GM r _{1} r _{2}}{r _{1} + r _{2}}$ (2) $0$ (3) $m \frac{2 GM ( r _{1} + r _{2} )}{r _{1} r _{2}}$ (4) $\frac{2 GMm r _{1}}{( r _{1} + r _{2} ) r _{2}}$ 254. $Aman o f 50 k g mass is standing in a gravity free space at a height of 10 m above the floor. He throws a stone of 0.5 kg mass downwards with a speed 2 m / s . When the stone reaches the floor, the distance of the man above the floor will be$ (1) $9.9 m$ (2) $10.1 m$ (3) $10 m$ (4) $20 m$ 255. $A small point of mass m is placed at a distance 2 R from the centre ' O' of a big uniform solid sphere of mass M and radius R. The gravitational force on ' m ' due to M is F_{1} . A spherical part of radius R /3 is removed from the big sphere as shown in the figure and the gravitational force on m due to remaining part of M is found to be F_{2} . The value of ratio F_{1} : F_{2} is$ (1) $12 : 11$ (2) $11 : 10$ (3) $12 : 9$ (4) $16 : 9$ 256. $Four spheres each of mass m form a square of side d (as shown in figure). A fifth sphere of mass M is situated at the center of square. The total gravitational potential energy of the system is:$ (1) $- \frac{G m}{d} [(4 + 2) m + 42 M]$ (2) $- \frac{G m}{d} [(4 + 2) m + 42 m]$ (3) $- \frac{G m}{d} [3 m^{2} + 42 M]$ (4) $- \frac{G m}{d} [6 m^{2} + 42 M]$ 257. $The kinetic energies of a planet in an elliptical orbit about the Sun, at positions A, B and C are K_{A}, K_{B} and K_{C}, respectively. AC is the major axis and SB is perpendicular to AB at the position of the Sun S as shown in the figure. Then$ (1) $K_{A} < K_{B} < K_{C}$ (2) $K_{B} > K_{A} > K_{C}$ (3) $K_{B} < K_{A} < K_{C}$ (4) $K_{A} > K_{B} > K_{C}$ 258. $A body of mass m is placed on earth’s surface which is taken from earth surface to a height of h = 3 R, then change in gravitational potential energy is$ (1) $\frac{m g R}{4}$ (2) $\frac{2}{3} m g R$ (3) $\frac{3}{4} m g R$ (4) $\frac{m g R}{2}$ 259. $A thin uniform annular disc (see figure) of mass M has outer radius 4R and inner radius 3R. The work required to take a unit mass from point P on its axis to infinity is:$ (1) $\frac{2 GM}{7 R} (42 - 5)$ (2) $- \frac{2 GM}{7 R} (42 - 5)$ (3) $\frac{GM}{4 R}$ (4) $\frac{2 GM}{5 R} (2 - 1)$ 260. $A body of mass m is projected with velocity λ ve in vertically upward direction from the surface of the earth into space. It is given that ve is escape velocity and λ < 1. If air resistance is considered to the negligible, then the maximum height from the center of earth, to which the body can go, will be (R: radius of earth)$ (1) $\frac{R}{1 + λ ^{2}}$ (2) $\frac{R}{1 - λ ^{2}}$ (3) $\frac{R}{1 - λ}$ (4) $\frac{λ ^{2} R}{1 - λ ^{2}}$ 261. $The weight of a body on the earth is 400 N. Then weight of the body when taken to a depth half of the radius of the earth will be:$ (1) $200 N$ (2) $Zero$ (3) $100 N$ (4) $300 N$ 262. $Infinite number of bodies, each of mass 2 kg are situated on x-axis at distances 1m, 2m, 4m, 8m, ........ respectively, from the origin. The resulting gravitational potential due to this system at the origin will be:$ (1) $- \frac{8}{3} G$ (2) $- \frac{4}{3} G$ (3) $-4G$ (4) $-G$ 263. $Two identical particles each of mass m go round a circle of radius a under the action of their mutual gravitational attraction. The angular speed of each particle will be:$ (1) $\frac{Gm}{a ^{3}}$ (2) $\frac{Gm}{8 a ^{3}}$ (3) $\frac{Gm}{4 a ^{3}}$ (4) $\frac{Gm}{2 a ^{3}}$ 264. $Assuming the earth to be a sphere of uniform mass density, the weight of a body at a depth d = \frac{R}{2} from the surface of earth, if its weight on the surface of earth is 200 N, will be: (Given R = radius of earth)$ (1) $300 N$ (2) $100 N$ (3) $400 N$ (4) $500 N$ 265. $A particle of mass M is situated at the center of a spherical shell of same mass and radius a. The gravitational potential at a point situated at \frac{a}{2} distance from the center, will be$ (1) $- \frac{3 GM}{a}$ (2) $- \frac{2 GM}{a}$ (3) $- \frac{GM}{a}$ (4) $- \frac{4 GM}{a}$ 266. $Two satellites A and B move round the earth in the same orbit. The mass of A is twice the mass of B. The quantity which is same for the two satellites will be$ (1) $Speed$ (2) $Kinetic energy$ (3) $Total energy$ (4) $Potential energy$ 267. $The work done to raise a mass m from the surface of the earth to a height h, which is equal to the radius of the earth, is [NEET 2019]$ (1) $2 mgR$ (2) $\frac{1}{2} mgR$ (3) $\frac{3}{2} mgR$ (4) $mgR$ 268. $A planet having mass 9 M_{e} and radius 4 R_{e}, where M_{e} and R_{e} are mass and radius of earth respectively, has escape velocity in km s^{- 1} given by: (Given escape velocity on earth V_{e} = 11. 2 x 1 0^{3} m s^{- 1})$ (1) $67.2$ (2) $16.8$ (3) $11.2$ (4) $33.6$ 269. $In the reported figure of earth, the value of acceleration due to gravity is same at point A and C but it is smaller than that of its value at point B (surface of the earth). The value of OA : AB will be : y . The value of x is$ 270. $The initial velocity v_{i} required to project a body vertically upward from the surface of the earth to reach a height of 10 R, where R is the radius of the earth, may be described in terms of escape velocity v_{e} such that v_{i} = \frac{x}{y} \times v_{e} . The value of x will be$