TOPIC – GRAVITATION Prepared by Sombir Ahlawat Sir Mob. No.
-- 8930818212
EXAMPLE in fig. The two spheres are held fixed. A projectile of
mass m is projected from the surface of the sphere of
1) Let the speed of the planet at the perihelion P in fig. mass M directly towards the centre of the second sphere.
be vP and the Sun-planet distance SP be rp. Relate {rp. vp} Obtain an expression for the minimum speed v of the
to the corresponding quantities at the aphelion {rA . vA}. projectile so that it reaches the surface of the second
will the planet take equal times to traverse BAC and sphere.
CPB?
2) Three equal masses of
m kg each are fixed at
the vertices of an
equilateral triangle ABC.
a) What is the force 5) The planet Mars has two moons, phobos and delmos.
acting on a mass 2 m (i) phobos has a period 7 hours , 39 minutes and an
placed at the centroid G orbital radius of 9.4 103 km. Calculate the mass of mars.
of the triangle? (ii) Assume that earth and mars move in circular orbits
b) What is the force if around the sun, with the martian orbit being 1.52 times
the mass at the vertex A is double? the orbital radius of the earth. What is the length of the
Take AG = BG = CG = 1 m (see fig.) martian year in days?
3) Find the potential energy of a system of four particles 6) Weighing the Earth: You are given the following data:
placed at the vertices of a square of side l. Also obtain g = 9.81 ms-2, RE = 6.37 106 m, the distance to the moon
the potential at the centre of the square. R = 3.84 108 m and the time period of the moon’s
revolution is 27.3 days. Obtain the mass of the Earth ME
in two different ways.
4) Two uniform solid spheres of equal radii R, but mass M
and 4 M have a centre to centre separation 6 R, as shown
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7) A 400 kg satellite is in a circular orbit of radius 2RE c) Acceleration due to gravity is independent of mass of
about the Earth. How much energy is required to transfer the earth/mass of the body.
it to a circular orbit of radius 4RE? What are the changes d) The formula – G Mm(1/r2 – 1/r1) is more /less accurate
in the kinetic and potential energies? than the formula mg (r2 – r1) for the difference of
potential energy between two points r2 and r1 distance
away from the centre of the earth.
EXERCISES
3) Suppose there existed a planet that went around the
1) Answer the following sun twice as fast as the earth. What would be its orbital
a) You can shield a charge from electrical forces by size as compared to that of the earth?
putting it inside a hollow conductor. Can you shield a
body from the gravitational influence of nearby matter
by putting it inside a hollow sphere or by some other
means?
b) An astronaut inside a small space ship orbiting around
the earth cannot detect gravity. If the space station
orbiting around the earth has a large size, can he hope to
detect gravity? 4) Io, one of the satellites of Jupiter, has an orbital period
of 1.769 days and the radius of the orbit is 4.22 108 m.
c) If you compare the gravitational force on the earth due Show that the mass of Jupiter is about one-thousandth
to the sun to that due to the moon, you would find that that of the sun.
the Sun’s pull is greater than the moon’s pull. (you can
check this yourself using the data available in the
succeeding exercises). However, the tidal effect of the
moon’s pull is greater than the tidal effect of sun. why?
5) Let us assume that our galaxy consists of 2.5 1011
stars each of one solar mass. How long will a star at a
distance of 50,000 ly from the galactic centre take to
complete one revolution? Take the diameter of the Milky
Way to be 105 ly.
2) Choose the correct alternative: 6) Choose the correct alternative;
a) Acceleration due to gravity increases/decreases with a) If the zero of potential energy is at infinity, the total
increasing altitude. energy of an orbiting satellite is negative of its
b) Acceleration due to gravity increases/ decreases with kinetic/potential energy.
increasing depth (assume the earth to be a sphere of b) The energy required to launch an orbiting satellite out
uniform density). of earth’s gravitational influence is more/less than the
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energy required to project a stationary object at the mass density has the direction indicated by the arrow
same height (as the satellite) out of earth’s influence. (fig) (i) a, (ii) b, (iii) c, (iv) 0.
11) For the above problem, the direction of the
7) Does the escape speed of a body from the earth gravitational intensity at an arbitrary point P is indicated
depend on (a) the mass of the body, (b) the location from by the arrow (i) d, e, (iii) f, (iv) g.
where it is projected, (c) the direction of projection, (d)
the height of the location from where the body is
launched ?
12) A rocket is fired from the earth towards the sun. At
8) A comet orbits the sun in a highly elliptical orbit. Does what distance from the earth’s centre is the gravitational
the comet have a constant (a) linear speed, (b) angular force on the rocket zero? Mass of the sun = 2 1030 kg,
speed, (c) angular momentum, (d) kinetic energy, (e) mass of the earth = 6 1024 kg. Neglect the effect of
potential energy, (f) total energy throughout its orbit? other planets etc. (orbital radius = 1.5 1011 m).
Neglect any mass loss of the comet when it comes very
close to the Sun.
13) How will you ‘weigh the sun’, that is estimate its
9) Which of the following symptoms is likely to afflict an mass? The mean orbital radius of the earth around the
astronaut in space (a) swollen feet, (b) swollen face, (c) sun is 1.5 108 km.
headache, (d) orientational problem.
10) In the
following two
exercises, choose 14) A Saturn year is 29.5 times the earth year. How far is
the correct answer the Saturn from the sun if the earth is 1.50 108 km away
from among the from the sun?
given ones:
The gravitational
intensity at the centre of a hemispherical shell of uniform
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19) A satellite orbits the earth at a height of 400 km
above the surface. How much energy must be expended
15) A body weighs 63 N on the surface of the earth. What to rocket the satellite out of the earth’s gravitational
is the gravitational force on it due to the earth at a height influence? Mass of the satellite = 200 kg; mass of the
equal to half the radius of the earth? earth’s = 6.0 1024 kg; radius of the earth = 6.4 106 m; G
= 6.67 10-11 N m2 kg-2.
16) Assuming the earth to be a sphere of uniform mass
density, how much would a body weigh half way down 20) Two starts each of one solar mass (=2 1030 kg) are
to the centre of the earth if it weighed 250 N on the approaching each other for a head on collision. When
surface? they are a distance 109 km, their speeds are negligible.
What is the speed with which they collide? The radius of
each star s 104 km. Assume the stars to remain
undistorted until they collide. (use the known value of
G).
17) A rocket is fired vertically with a speed of 5 km s-1
from the earth’s surface. How far from the earth does
the rocket go before returning to the earth? Mass of the
earth = 6.0 1024 kg; mean radius of the earth’s = 6.4
106 m; G = 6.67 10-11 N m2 kg-2. 21) Two heavy spheres each of mass 100kg and radius
0.10 m are placed 1.0 m apart on a horizontal table.
What is the gravitational force and potential at the mid
point of the line joining the centres of the spheres? Is an
object at that point in equilibrium? If so, is the
equilibrium stable or unstable?
18) The escape speed of a projectile on the earth’s
surface is 11.2 km s-1. A body is projected out with thrice
this speed . What is the speed of the body far away from
the earth? Ignore the presence of the sun and other
planets.
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