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 ball 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 K and natural length R, the value of K is (R is radius of Earth and M mass of Earth)

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 :–

A solid sphere of mass M and radius R is surrounded by a spherical shell of same mass M and radius 2R as shown. A small particle of mass m is released from rest from a height h (<<R) above the shell. There is a hole in the shell With what approximate speed will it collide at B?

A solid sphere of mass M and radius R is surrounded by a spherical shell of same mass M and radius 2R as shown. A small particle of mass m is released from rest from a height h (<<R) above the shell. There is a hole in the shell What time will it take to move from A to B?

A solid sphere of mass M and radius R is surrounded by a spherical shell of same mass M and radius 2R as shown. A small particle of mass m is released from rest from a height h (<<R) above the shell. There is a hole in the shell In what time will it enter the hole at A:–

A triple star system consists of two stars, each of mass m, in the same circular orbit about central star with mass . The two outer stars always lie at opposite ends of a diameter of their common circular orbit. The radius of the circular orbit is r = m, and the orbital period of each star is s. [Take = 10 and G = ] The total mechanical energy of the system is

A triple star system consists of two stars, each of mass m, in the same circular orbit about central star with mass . The two outer stars always lie at opposite ends of a diameter of their common circular orbit. The radius of the circular orbit is r = m, and the orbital period of each star is s. [Take = 10 and G = ] The orbital velocity of each star is

A triple star system consists of two stars, each of mass m, in the same circular orbit about central star with mass . The two outer stars always lie at opposite ends of a diameter of their common circular orbit. The radius of the circular orbit is r = m, and the orbital period of each star is s. [Take = 10 and G = ] The mass m of the outer stars is

A planet is revolving around the sun is an elliptical orbit as shown in figure. Select correct alternative(s)

If the law of gravitation be such that the force of attraction between two particles vary inversely as the power of their separation, then the graph of orbital velocity v₀ plotted against the distance r of a satellite from the earth's centre on a log-log scale is shown alongside. The slope of line will be-

A particle is projected from point A, that is at a distance 4R from the centre of the Earth, with speed in a direction making with the line joining the centre of the Earth and point A, as shown. Find the speed of particle (in m/s) if particle passes grazing the surface of the earth. Consider gravitational interaction only between these two.

The figure shows elliptical orbit of a planet m about the sun S. The shaded area SCD is twice the shaded area SAB. If be the time for the planet to move from C to D and is the time to move from A to B, then :

Velocity of efflux in Torricelli's theorem is given by , here h is the height of hole from the top surface, after that, motion of liquid can be treated as projectile motion. Liquid is filled in a vessel of square base up to a height of 2m as shown in figure (i). In figure (ii) the vessel is tilted from horizontal at What is the velocity of efflux in this case. Liquid does not spills out?

The number of values of c such that the straight line y = 4x + c touches the curve + y² = 1 is

If P(x, y), F₁=(3,0), F₂ = (– 3, 0)and16x² + 25 y² = 400, then P F₁ + P F₂ =

Let P be a variable point on the ellipse + = 1 with foci F₁ and F₂. If A is the area of the triangle PF₁ F₂, then the maximum value of A is-

Area of triangle with given vertices can be calculated using the matrix determimnant method.