Gravitation

An asteroid is moving directly towards the
centre of the earth. When at a distance of
10R (R is the radius of the earth) from the earths
centre, it has a speed of 12 km/s. Neglecting
the effect of earths atmosphere, what will be the
speed o...

A satellite of mass m is launched vertically upwards with an initial speed u from the surface of the earth. After it reaches height R (R = radius of the earth), it ejects a rocket
of mass \({m \over {10}}\)
so that subsequently the
satellit...

A box weight 196 N on a spring balance at the north pole. Its weight recorded on the same
balance if it is shifted to the equator is close to (Take g = 10 ms–2 at the north pole and the radius
of the earth = 6400 km) :

A satellite is in an elliptical orbit around a planet P. It is observed that the velocity of the satellite
when it is farthest from the planet is 6 times less than that when it is closest to the planet. The
ratio of distances between the sa...

Two planets have masses M and 16 M and their radii are \(a\) and 2\(a\), respectively. The separation
between the centres of the planets is 10\(a\). A body of mass m is fired from the surface of the larger
planet towards the smaller planet ...

The value of the acceleration due to gravity is
g1 at a height h = \({R \over 2}\) (R = radius of the earth) from the surface of the earth. It is again equal
to g1 at a depth d below the surface of the
earth. The ratio $$\left( {{d \over R}...

The acceleration due to gravity on the earth’s
surface at the poles is g and angular velocity of
the earth about the axis passing through the
pole is \(\omega\). An object is weighed at the equator
and at a height h above the poles by usin...

On the x-axis and at a distance x from the origin, the gravitational field due a mass distribution is
given by \({{Ax} \over {{{\left( {{x^2} + {a^2}} \right)}^{3/2}}}}\) in the x-direction. The magnitude of gravitational potential on the x...

A body is moving in a low circular orbit about a planet of mass M and radius R. The radius of the
orbit can be taken to be R itself. Then the ratio of the speed of this body in the orbit to the escape
velocity from the planet is:

A satellite is moving in a low nearly circular orbit around the earth. Its radius is roughly equal to
that of the earth’s radius Re
. By firing rockets attached to it, its speed is instantaneously increased
in the direction of its motion so...

The mass density of a planet of radius R varies with the distance r from its centre as
\(\rho\)(r) = \({\rho _0}\left( {1 - {{{r^2}} \over {{R^2}}}} \right)\).
Then the gravitational field is maximum at :

The mass density of a spherical galaxy varies
as
\({K \over r}\) over a large distance ‘r’ from its centre.
In that region, a small star is in a circular orbit
of radius R. Then the period of revolution, T
depends on R as :

The height ‘h’ at which the weight of a body will
be the same as that at the same depth ‘h’ from
the surface of the earth is (Radius of the earth
is R and effect of the rotation of the earth is
neglected)

The energy required to take a satellite to a height 'h' above Earth surface (radius of Earth = 6.4 \(\times\) 103 km) is E1 and kinetic energy required for the satellite to be in a circular orbit at this height is E2. The value of h for w...

A solid sphere of mass 'M' and radius 'a' is
surrounded by a uniform concentric spherical
shell of thickness 2a and mass 2M. The
gravitational field at distance '3a' from the
centre will be :

A test particle is moving in a circular orbit in
the gravitational field produced by a mass
density \(\rho (r) = {K \over {{r^2}}}\) . Identify the correct relation
between the radius R of the particle's orbit and
its period T

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 other's gravitational field
in a circular orbit circumscribing the square?

A rocket has to be launched from earth in such
a way that it never returns. If E is the minimum
energy delivered by the rocket launcher, what
should be the minimum energy that the
launcher should have if the same rocket is to
be launched fr...

A satellite of mass M is in a circular orbit of radius R about the centre of the earth. A meteorite of the same mass, falling towards the earth, collides with the satellite completely inelastically. The speeds of the satellite and the meteo...

A straight rod of length L extends from x = a to x = L + a. The gravitational force it exerts on a point mass 'm' at x = 0, if the mass per unit length of the rod is A + Bx2 , is given by :

Two satellites, A and B, have masses m and 2m respectively. A is in a circular orbit of radius R, and B is in a circular orbit of radius 2R around the earth. The ratio of their kinetic energies, TA/TB, is ;

The ratio of the weights of a body on the Earth’s surface to that on the surface of a planets is 9 : 4. The mass
of the planet is
\({1 \over 9}\)
th of that of the Earth. If 'R' is the radius of the Earth, what is the radius of the planet ?...

A satellite is revolving in a circular orbit at a height h form the earth surface, such that h < < R where R is the earth. Assuming that the effect of earth's atmosphere can be neglected the minimum increase in the speed required so that th...

A satellite is moving with a constant speed v in circular orbit around the earth. An object of mass ‘m’ is ejected from the satellite such that it just escapes from the gravitational pull of the earth. At the time of ejection, the kinetic e...

Two stars of masses 3 \(\times\) 1031 kg each, and at distance 2 \(\times\) 1011 m rotate in a plane about their common centre of mass O. A meteorite passes through O moving perpendicular to the star’s rotation plane. In order to escape...

The value of acceleration due to gravity at
Earth's surface is 9.8 ms–2. The altitude above
its surface at which the acceleration due to
gravity decreases to 4.9 ms–2, is close to :
(Radius of earth = 6.4 × 106 m)

A spaceship orbits around a planet at a height of 20 km from its surface. Assuming that only gravitational
field of the planet acts on the spaceship, what will be the number of complete revolutions made by the
spaceship in 24 hours around t...

Suppose that the angular velocity of rotation of earth is increased. Then, as a consequence :

A body of mass m is moving in a circular orbit of radius R about a planet of mass M. At some instant, it splits into two equal masses. The first mass moves in a circular orbit of radius \({R \over 2},\) and the other mass, in a circular orb...

Take the mean distance of the moon and the sun from the earth to be \(0.4 \times {10^6}\) km and \(150 \times {10^6}\) km respectively. Their masses are \(8 \times {10^{22}}\) kg and \(2 \times {10^{30}}\) kg respectively. The radius of the...

A particle is moving with a uniform speed in a circular orbit of radius R in a central force inversely
proportional to the nth power of R. If the period of rotation of the particle is T, then :

The mass density of a spherical body is given by
\(\rho\) (r) = \({k \over r}\) for r \(\le\) R and \(\rho\) (r) = 0 for r > R,
where r is the distance from the centre.
The correct graph that describes qualitatively the acceleration, a...

If the Earth has no rotational motion, the weight of a person on the equator is W. Determine the speed with which the earth would have to rotate about its axis so that the person at the equator will weigh \({3 \over 4}\) W. Radius of the E...

The variation of acceleration due to gravity \(g\) with distance d from centre of the earth is best represented by
(R = Earth’s radius):

Figure shows elliptical path abcd of a planet around the sun S such that the area of triangle csa is \({1 \over 4}\) the area of the ellipse. (See figure) With db as the semimajor axis, and ca as the semiminor axis. If t1 is the time taken...

An astronaut of mass m is working on a satellite orbiting the earth at a distance h from the earth’s surface. The radius of the earth is R, while its mass is M. The gravitational pull FG on the astronaut is :

A satellite is revolving in a circular orbit at a height \('h'\) from the earth's surface (radius of earth \(R;h < < R\)). The minimum increase in its orbital velocity required, so that the satellite could escape from the earth's gravitat...

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 i...

Four particles, each of mass \(M\) and equidistant from each other, move along a circle of radius \(R\) under the action of their mutual gravitational attraction. The speed of each particle is :

What is the minimum energy required to launch a satellite of mass \(m\) from the surface of a planet of mass \(M\) and radius \(R\) in a circular orbit at an altitude of \(2R\)?

The mass of a spaceship is \(1000\) \(kg.\) It is to be launched from the earth's surface out into free space. The value of \(g\) and \(R\) (radius of earth ) are \(10\,m/{s^2}\) and \(6400\) \(km\) respectively. The required energy for thi...

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:

The height at which the acceleration due to gravity becomes \({g \over 9}\) (where \(g=\) the acceleration due to gravity on the surface of the earth) in terms of \(R,\) the radius of the earth, is:

A planet in a distant solar system is \(10\) times more massive than the earth and its radius is \(10\) times smaller. Given that the escape velocity from the earth is \(11\,\,km\,{s^{ - 1}},\) the escape velocity from the surface of the p...

This question contains Statement - \(1\) and Statement - \(2\). of the four choices given after the statements, choose the one that best describes the two statements.
Statement - \(1\):
For a mass \(M\) kept at the center of a cube of side ...

If \({g_E}\) and \({g_M}\) are the accelerations due to gravity on the surfaces of the earth and the moon respectively and if Millikan's oil drop experiment could be performed on the two surfaces, one will find the ratio
$${{electro\,\,ch...

A particle of mass \(10\) \(g\) is kept on the surface of a uniform sphere of mass \(100\) \(kg\) and radius \(10\) \(cm.\) Find the work to be done against the gravitational force between them to take the particle far away from the sphere ...

The change in the value of \(g\) at a height \(h\) above the surface of the earth is the same as at a depth \(d\) below the surface of earth. When both \(d\) and \(h\) are much smaller than the radius of earth, then which one of the followi...

Average density of the earth

Suppose the gravitational force varies inversely as the nth power of distance. Then the time period of a planet in circular orbit of radius \(R\) around the sun will be proportional to