Gravitation PYQs Set 10 (PDF)

Set 10 of 12 · Questions 271-300 of 351

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Mere Teacher · Previous Year Questions JEE Main Physics · Gravitation · Set 10 (Q271-300)
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271. $The bodies of masses 100 kg and 8100 kg are held at a distance of 1 m . The gravitational field at a point on the line joining them is zero. The gravitational potential at that point in J / kg is (G = 6.67 \times 10^{- 11} Nm^{2} / kg^{2})$ (1) $- 6.67 \times 10^{- 7}$ (2) $- 6.67 \times 10^{- 10}$ (3) $- 13.34 \times 10^{- 7}$ (4) $- 6.67 \times 10^{- 9}$ 272. $A body is projected vertically upwards from the surface of earth with a velocity equal to one third of escape velocity. The maximum height attained by the body will be: (Take radius of earth = 6400 km and g=10 m s^{- 2})$ (1) $800km$ (2) $1600km$ (3) $2133 km$ (4) $4800 km$ 273. $The earth (mass = 6 \times 1 0^{24} k g) revolves around the sun with an angular velocity of 2 \times 1 0^{- 7} r a d / s in a circular orbit of radius 1.5 \times 1 0^{8} km. The force exerted by the sun on the earth, in newton, is$ (1) $36 \times 1 0^{21}$ (2) $27 \times 1 0^{39}$ (3) $zero$ (4) $18 \times 1 0^{25}$ 274. $Four identical particles of mass m are kept at the four corners of a square. If the gravitational force exerted on one of the masses by the other masses is (\frac{2 2 + 1}{32}) \frac{Gm ^{2}}{L ^{2}}, the length of the sides of the square is$ (1) $\frac{L}{2}$ (2) $4 L$ (3) $3 L$ (4) $2 L$ 275. $A spherical body of radius R consists of a fluid of constant density and is in equilibrium under its own gravity., If P(r) is the pressure at r (r < R), then the correct option(s) is (are)$ (1) $P (r = 0) = 0$ (2) $\frac{P ( r = 3 R /4 )}{P ( r = 2 R /3 )} = \frac{63}{80}$ (3) $\frac{P ( r = 3 R /5 )}{P ( r = 2 R /5 )} = \frac{16}{21}$ (4) $\frac{P ( r = R /2 )}{P ( r = R /3 )} = \frac{20}{27}$ 276. $Two bodies of masses m and 4 m are placed at a distance r. The gravitational potential at a point on the line joining them where the gravitational field is zero is:$ (1) $zero$ (2) $- \frac{4 Gm}{r}$ (3) $- \frac{6 Gm}{r}$ (4) $- \frac{9 Gm}{r}$ 277. $A solid sphere of radius R gravitationally attracts a particle placed at R from its centre with a force F_{1} . Now a spherical cavity of radius (\frac{R}{2}) is made in the sphere (as shown in figure) and the force becomes F_{2} .. The value of F_{1} : F_{2} is$ (1) $41 : 50$ (2) $36 : 25$ (3) $50 : 41$ (4) $25 : 36$ 278. $Assume that the earth is a solid sphere of uniform density and a tunnel is dug along its diameter throughout the earth. It is found that when a particle is released in this tunnel, it executes a simple harmonic motion. The mass of the particle is 100 g. The time period of the motion of the particle will be (approximately) (take g = 10 m s^{- 2}, radius of earth = 6400 km)$ (1) $24 hours$ (2) $1 hour 24 minutes$ (3) $1 hour 40 minutes$ (4) $12 hours$ 279. $Kepler's second law (law of areas) is nothing but a statement of$ (1) $work-energy theory$ (2) $conservation of linear momentum$ (3) $conservation of angular momentum$ (4) $conservation of energy$ 280. $A ring has a mass M and radius R . The distance of the point on its geometric axis from its center at which the gravitational field is strongest is$ (1) $\frac{R}{2}$ (2) $\frac{R}{4}$ (3) $\frac{R}{3}$ (4) $\frac{R}{2}$ 281. $A planet is revolving around the sun in which of the following is correct statement?$ (1) $The time taken in travelling DAB is less than that for BCD.$ (2) $The time taken in travelling DAB is greater than that for BCD.$ (3) $The time taken in travelling CDA is less than that for ABC.$ (4) $The time taken in travelling CDA is greater than that for ABC$ 282. $From a solid sphere of mass M and radius R, a spherical portion of radius R/2 is removed, as shown in the figure. Taking gravitational potential V = 0 at r = \infty, the potential at the center of the cavity thus formed is :(G = gravitational constant)$ (1) $\frac{- GM}{2 R}$ (2) $\frac{- GM}{R}$ (3) $\frac{- 2 GM}{3 R}$ (4) $\frac{- 2 GM}{R}$ 283. $Two spherical planets P and Q have the same uniform density ρ, masses M_{P} and M_{Q}, with surface areas A and 4A, respectively. A spherical planet R also has uniform density ρ and its mass is (M_{P} + M_{Q}). The escape velocities from the planets P, Q and R, are V_{P}, V_{Q} and V_{R} respectively. Then$ (1) $V_{Q} > V_{R} > V_{P}$ (2) $V_{R} > V_{Q} > V_{P}$ (3) $V_{R} / V_{P} = 3$ (4) $V_{P} / V_{Q} = \frac{1}{2}$ 284. $The radius of a planet is twice the radius of earth. Both have almost equal average mass‐ densities. V_{P} and V_{E} are escape velocities of the planet and the earth, respectively, then$ (1) $V_{P} = 1.5 V_{E}$ (2) $V_{P} = 2 V_{E}$ (3) $V_{E} = 3 V_{P}$ (4) $V_{E} = 1.5 V_{P}$ 285. $A metal wire of uniform mass density having length L and mass M is bent to form a semicircular arc and a particle of mass m is placed at the centre of the arc. The gravitational force on the particle by the wire is:$ (1) $\frac{GmM π ^{2}}{L ^{2}}$ (2) $\frac{GMmπ}{2 L ^{2}}$ (3) $0$ (4) $\frac{2 GmMπ}{L ^{2}}$ 286. $The planet Mars has two moons, if one of them has a period 7 hours, 30 minutes and an orbital radius of 9. 0 x 1 0^{3} km. Find the mass of Mars. {Given \frac{4 π ^{2}}{G} = 6 \times 1 0^{11} N^{- 1} m^{- 2} kg^{2}}$ (1) $5.96 \times 1 0^{19} k g$ (2) $3.25 \times 1 0^{21} k g$ (3) $7.02 \times 1 0^{25} k g$ (4) $6.00 \times 1 0^{23} k g$ 287. $Two spherical bodies of masses M and 5 M and radii R and 2 R are released in free space with initial separation between their centres equal to 12 R . If they attract each other due to gravitational force only, then the distance covered by the smaller body before collision is$ (1) $2.5 R$ (2) $4.5 R$ (3) $7.5 R$ (4) $1.5 R$ 288. $A space ship of mass 2 x 1 0^{4} kg is launched into a circular orbit close to the earth surface. The additional velocity to be imparted to the space ship in the orbit to overcome the gravitational pull will be (if g = 10 m s^{- 2} and radius of earth = 6400 km):$ (1) $11.2 (2 - 1) km s^{- 1}$ (2) $8 (2 - 1) km s^{- 1}$ (3) $7.9 (2 - 1) km s^{- 1}$ (4) $7.4 (2 - 1) km s^{- 1}$ 289. $The escape velocity of a body on a planet 'A' is 12 k m s^{- 1} . The escape velocity of the body on another planet 'B', whose density is four times and radius is half of the planet 'A', is:$ (1) $12 k m s^{- 1}$ (2) $24 k m s^{- 1}$ (3) $36 k m s^{- 1}$ (4) $6 k m s^{- 1}$ 290. $A solid sphere of mass ’ M ’ and radius ’ a ’ is surrounded by a uniform concentric spherical shell of thickness 2 M and mass 2 M . The gravitational field at distance ’ 3 a ’ from the center will be:$ (1) $\frac{2 GM}{9 a ^{2}}$ (2) $\frac{GM}{9 a ^{2}}$ (3) $\frac{GM}{3 a ^{2}}$ (4) $\frac{2 GM}{3 a ^{2}}$ 291. $The kinetic energies of a planet in an elliptical orbit about the Sun, at positions A, B and C are K_{A}, K_{B} and K_{C}, respectively. AC is the major axis and SB is perpendicular to AC at the position of the Sun S as shown in the figure? Then$ (1) $K_{A} < K_{B} < K_{C}$ (2) $K_{A} > K_{B} > K_{C}$ (3) $K_{B} < K_{A} < K_{C}$ (4) $K_{B} > K_{A} > K_{C}$ 292. $A particle of mass m is thrown upwards from the surface of the earth, with a velocity u . The mass and the radius of the earth are, respectively, M and R. G is gravitational constant and g is acceleration due to gravity on the surface of the earth. The minimum value of u so that the particle does not return back to earth, is$ (1) $\frac{2 GM}{R ^{2}}$ (2) $\frac{2 GM}{R}$ (3) $\frac{2 g M}{R ^{2}}$ (4) $2 g R^{2}$ 293. $If the mass of the sun were ten times smaller and the universal gravitational constant were ten times larger in magnitude, which of the following is not correct? [NEET 2018]$ (1) $Time period of a simple pendulum on the earth would decrease$ (2) $Walking on the ground would become more difficult$ (3) $Raindrops will fall faster$ (4) $g on the earth will not change$ 294. $Find the gravitational force of attraction between the ring and sphere as shown in the diagram, where the plane of the ring is perpendicular to the line joining the centers. If 8 R is the distance between the centres of a ring (of mass ’ m ’) and a sphere (mass ’ M ’) where both have equal radius ’ R ’ .$ (1) $\frac{8}{9} \cdot \frac{GmM}{R}$ (2) $\frac{8}{27} \cdot \frac{GmM}{R ^{2}}$ (3) $\frac{2 2}{3} \cdot \frac{GMm}{R ^{2}}$ (4) $\frac{1}{3 8} \cdot \frac{GMm}{R ^{2}}$ 295. $At what distance above and below the surface of the earth a body will have same weight, (take radius of earth as R.)$ (1) $5 R - R$ (2) $\frac{3 R - R}{2}$ (3) $\frac{R}{2}$ (4) $\frac{5 R - R}{2}$ 296. $Four identical particles of mass M are located at the corners of a square of side ’ a ’ . What should be their speed if each of them revolves under the influence of others' gravitational field in a circular orbit circumscribing the square?$ (1) $1.35 \frac{GM}{a}$ (2) $1.16 \frac{GM}{a}$ (3) $1.21 \frac{GM}{a}$ (4) $1.41 \frac{GM}{a}$ 297. $Two particles of equal mass ‘m’ go around a circle of radius R under the action of their mutual gravitational attraction. The speed of each particle with respect to their center of mass is:$ (1) $\frac{Gm}{4 R}$ (2) $\frac{Gm}{3 R}$ (3) $\frac{Gm}{2 R}$ (4) $\frac{Gm}{R}$ 298. $Consider two solid spheres of radii R_{1} = 1m, R_{2} = 2m and masses M_{1} and M_{2}, respectively. The gravitational field due to sphere (a) and (b) are shown. The value of \frac{M _{1}}{M _{2}} is:$ (1) $\frac{2}{3}$ (2) $\frac{1}{3}$ (3) $\frac{1}{2}$ (4) $\frac{1}{6}$ 299. $The largest and the shortest distance of the earth from the sun are r_{1} and r_{2} . Its distance from the sun when it is at perpendicular to the major‐axis of the orbit drawn from the sun is$ (1) $\frac{r _{1} + r _{2}}{4}$ (2) $\frac{r _{1} + r _{2}}{r _{1} - r _{2}}$ (3) $\frac{2 r _{1} r _{2}}{r _{1} + r _{2}}$ (4) $\frac{r _{1} + r _{2}}{3}$ 300. $A satellite of mass \frac{M}{2} is revolving around earth in a circular orbit at a height of \frac{R}{3} from earth surface. The angular momentum of the satellite is M \frac{GMR}{x} . The value of x is, where M and R are the mass and radius of earth, respectively. (G is the gravitational constant)$