Three point masses, m each are at the corners of an equilateral triangle of side a. Their separations do not change when the system rotates about the centre of the triangle. For this, the time period of rotation must be proportional to :
The correct answers are: a3/2, m–1/2
A solid sphere of uniform density and radius 4units is located with its centre at the origin of coordinates, O. Two spheres of equal radii of 1unit, with their centres at A(-2,0,0) and B(2, 0, 0) respectively, are taken out of the solid sphere, leaving behind spherical cavities as shown in the figure.
Use arguments of symmetry as the yz plane divides the objects symmetrically.
The correct answers are: The gravitational force due to this object at the origin is zero, The gravitational potential is the same at all points of the circle y2 + z2 = 36, The gravitational potential is the same at all points of the circle y2 + z2 = 4
A satellite revolves around a planet in circular orbit of radius R (much larger than the radius of the planet) with a time period of revolution T. If the satellite is stopped and then released in its orbits (Assume that the satellite experiences gravitational force due to the planet only).
It will fall because mg is acting on it towards the centre of planet and initial velocity is zero. It’ll move in straight line.
By energy conservation
using this we get V = f(r)
R' = radius of the planet.
In the final expression (on in the beginning itself)
The correct answers are: It will fall on the planet, The time of fall of the satellite on the planet is nearly
Let V and E denotes the gravitational potential and gravitational field at the point. It is possible to have :
A) At ∞ both V and E are zero.
B) Let, V∞=GM/R for an unit mass.
So, VR=0 i.e. at the radius R of solid sphere(mass M) and ER=GM/R2
C) Inside a spherical shell V =GM/R and E =0
Thus, all the above are correct.
A small mass m is moved slowly from the surface of the earth to a height h above the surface. The work done (by an external agent) in doing this is :
For, h<<R force is constant and is equal to mg
Therefore, work done is mgh in moving by distance h
gravitational potential at x is − GM/x
Work done in moving from x=R to x=2R is − GMm/2R + GMm/R
which is equal to 1/2 mgR
A satellite close to the earth is in orbit above the equator with the period of rotation of 1.5hours. If it is above a point P on the equator at some time, it will be above P again after time :
Let ω0 = the angular velocity of the earth about its axis.
Let ω = the angular velocity of the satellite
For a satellite rotating from west to east (the same as the earth), the relative angular velocity, ω1 = ω - ω0
The period of rotation relative to the earth
For a satellite rotating from east to west (opposite to the earth), the relative angular velocity, ω2 = ω + ω0.
The correct answers are: 1.6hours if it is rotating from west to east, 24/17hours if it is rotating from east to west
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 :
F ∝ T, if r < R, and
if r > R.
The correct answers are:
A double star is a system of two stars of masses m and 2m, 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 :
The correct answers are: r3/2, m–1/2
A binary star is a system of two stars rotating about their centre of mass only under their mutual gravitational attraction. Let the stars have masses m and 2m and let their separation be l. Their time period of rotation about their centre of mass will be proportional to :
The correct answers are: l3/2, m–1/2
An object is weighed at the North Pole by a beam balance and a spring balance, giving readings of WB and WS respectively. It is again weighted in the same manner at the equator, giving readings of W′B and W′S respectively. Assume that the acceleration due to gravity is the same everywhere and that the balances are quite sensitive.