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A double star is a system of two stars of masses m and 2 m, rotating about their centre of mass only under their mutual gravitational attraction. If r is the separation between these two stars then their time period of rotation about their centre of mass will be proportional to :

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Important Questions on Gravitation

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A solid sphere of uniform density and radius 4 units is located with its centre at the origin O of coordinates. Two spheres of equal radii 1 unit, with their centres at A(-2,0,0) and B(2,0,0) respectively, are taken out of the solid leaving behind spherical cavities as shown in figure. Then :-

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The magnitudes of the gravitational field at distance r1 and r2 from the centre of a uniform, sphere of radius R and mass M are F1 and F2 respectively. then :-
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Mark the correct statement/s :-
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Gravitational potential at the centre of curvature of a hemispherical bowl of radius R and mass M is V :-
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Suppose a smooth tunnel is dug along a straight line joining two points on the surface of the Earth and a particle is dropped from rest at its one end. Assume that mass of Earth is uniformly distributed over lts volume. Then
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A small ball of mass ' m ' is released at a height ' R ' above the Earth surface, as shown in the figure. If the maximum depth of the hall to which it goes is R/2 inside the Earth through a narrow grove before coming to rest momentarily. The grove, contain an ideal spring of spring constant Kand natural length R, the value of Kis (R is radius of Earth and M mass of Earth)

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A small point object of mass m is transferred from the centre of the base of a uniform solid hemisphere of mass M and radius R to infinity. The work performed in the process by the gravitational force exerted on the particle by the hemisphere is
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Masses and radii of earth and moon are M1, M2 and R1, R2 respectively. The distance between their centre is d. The minimum velocity is given to mass M from the mid-point of the line joining their centre so that it will escape: