Chapter 8 - GRAVITATION

A satellite is in an elliptic orbit around the earth with
aphelion of 6R and perihelion of 2 R where R= 6400 km is
the radius of the earth. Find eccentricity of the orbit. Find
the velocity of the satellite at apogee and perigee. What
should be done if this satellite has to be transferred to a
circular orbit of radius 6R ?

Earth’s orbit is an ellipse with eccentricity 0.0167. Thus,
earth’s distance from the sun and speed as it moves around
the sun varies from day to day. This means that the length
of the solar day is not constant through the year. Assume
that earth’s spin axis is normal to its orbital plane and find
out the length of the shortest and the longest day. A day
should be taken from noon to noon. Does this explain
variation of length of the day during the year?

A satellite is to be placed in equatorial geostationary orbit around
earth for communication.
(a) Calculate height of such a satellite.
(b) Find out the minimum number of satellites that are needed to
cover entire earth so that at least one satellites is visible from
any point on the equator.

A star like the sun has several bodies moving around it at different
distances. Consider that all of them are moving in circular orbits.
Let r be the distance of the body from the centre of the star and let
its linear velocity be v, angular velocity w, kinetic energy K,
gravitational potential energy U, total energy E and angular
momentum l. As the radius r of the orbit increases, determine
which of the above quantities increase and which ones decrease.

Six point masses of mass m each are at the vertices of a regular
hexagon of side l. Calculate the force on any of the masses.

A star like the sun has several bodies moving around it at different
distances. Consider that all of them are moving in circular orbits.
Let r be the distance of the body from the centre of the star and let
its linear velocity be v, angular velocity w, kinetic energy K,
gravitational potential energy U, total energy E and angular
momentum l. As the radius r of the orbit increases, determine
which of the above quantities increase and which ones decrease.

An object of mass m is raised from the surface of the earth to a
height equal to the radius of the earth, that is, taken from a distance
R to 2R from the centre of the earth. What is the gain in its potential
energy?

Show the nature of the following graph for a satellite orbiting the
earth.
(a) KE vs orbital radius R
(b) PE vs orbital radius R
(c) TE vs orbital radius R.

Two identical heavy spheres are separated by a distance 10 times
their radius. Will an object placed at the mid point of the line joining
their centres be in stable equilibrium or unstable equilibrium?
Give reason for your answer.

Mean solar day is the time interval between two successive noon
when sun passes through zenith point (meridian).
Sidereal day is the time interval between two successive transit of
a distant star through the zenith point (meridian).
By drawing appropriate diagram showing earth’s spin and orbital
motion, show that mean solar day is four minutes longer than the
sidereal day. In other words, distant stars would rise 4 minutes
early every successive day.

What is the angle between the equatorial plane and the orbital
plane of
(a) Polar satellite?
(b) Geostationary satellite?

Out of aphelion and perihelion, where is the speed of the earth
more and why ?

The gravitational force between a hollow spherical shell (of radius
R and uniform density) and a point mass is F. Show the nature of
F vs r graph where r is the distance of the point from the centre of
the hollow spherical shell of uniform density.

An astronaut inside a small spaceship 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?

We can shield a charge from electric fields by putting it inside a
hollow conductor. Can we shield a body from the gravitational
influence of nearby matter by putting it inside a hollow sphere or
by some other means?

Is it possibe for a body to have inertia but no weight?

How is the gravitational force between two point masses affected
when they are dipped in water keeping the separation between
them the same?

What is the direction of areal velocity of the earth around the sun?

Draw areal velocity versus time graph for mars.

Give one example each of central force and non-central force.

Molecules in air in the atmosphere are attracted by gravitational
force of the earth. Explain why all of them do not fall into the earth
just like an apple falling from a tree.

Obtain on expression showing variation of acceleration due to gravity with
height?

How far away from the surface of earth does the value of g is reduced to 4% of
its value on the surface of the earth Given radius of earth = 6400km

A satellite is revolving is a circular path close to a planet of density P. find an
expression for its period of revolution?

If radius of earth is 6400km, what will be the weight of 1 quintal body if taken
to the height of 1600 km above the sea level?

Assuming earth to be a uniform sphere finds an expression for density of earth
in terms of g and G?

With two characteristics of gravitational force?

What is kepler’s law of periods? Show it mathematically?

Name two factors which determine whether a planet has atmosphere or not?

At what height above the surface of the earth value of acceleration due to
gravity is reduced to one fourth of its value on the surface of the earth?

Why is gravitational potential energy always negative?

Define Gravitational potential energy Hence deduces an expression for
gravitational potential energy of a body placed at a point sear the surface of
earth?

If T be the period of satellite revolving just above the surface of a planet

whose average density is p, show that PT² is a universal constant.

Show that value of g at a height h is same as the value of acceleration due of
gravity at a depth d = 2h

Determine the value of g at the bottom of an ocean 7km deep Given that

radius of earth is 6370 km and g = 9.8m|s².

The earths mass is 80 times that of moon and their diameters are in the ratio
4:1 respectively. What is the value of g on moon?

Find an expression for gravitational intensity due to earth at a point on its free
surface.

Find an expression for the weight of a body at the centre of the earth?

Where will a body weigh more at Delhi or at Shimla? Why?

A body is weightless at the centre of earth. Why?

The gravitational force between two blocks is F what would happen if a mass
of both the blocks as well as distance between them is doubled?

Find expressions for (1) potential energy (2) kinetic energy (3) total energy
for an artificial satellite.

Two uniform solid spheres of radii R and 2R are at rest with their surfaces just
touching. Find the force of gravitational attraction between them if density of
spheres be P?

A 400kg satellite in a circular orbit of radius 2 Re about the earth calculate the

kinetic energy potential energy and total energy of the satellite?

RE = 6.4x10⁶m

M = 6x10²⁴kg

Show that an artificial satellite circling round the earth in an orbit of radius
obeys keeper’s third low?

State two essential requisites of geostationary satellite?

The distance of the planet Jupiter from the sun is 5.2 times that of the earth.
Find the period of the Jupiter’s revolution around the sun?

Distance between two bodies is increased to three times its original value.
What is the effect on the gravitational force between them?

Which is greater the attraction of the earth for 1 kg of aluminum or aluminum
or attraction of 1kg of aluminum for the earth?

On which fundamental law of physics is keplers second law is based?

 A rocket is fired ‘vertically’ from the surface of mars with a speed of 2 km s–1. If 20% of its initial energy is lost due to Martian atmospheric resistance, how far will the rocket go from the surface of mars before returning to it? Mass of mars = 6.4× 1023 kg; radius of mars = 3395 km; G = 6.67× 10^-11 N m² kg^–2.

A spaceship is stationed on Mars. How much energy must be expended on the spaceship to launch it out of the solar system? Mass of the space ship = 1000 kg; mass of the Sun = 2 × 10³⁰ kg; mass of mars = 6.4 × 10²³ kg; radius of mars = 3395 km; radius of the orbit of mars = 2.28 × 10⁸kg; G= 6.67 × 10^–11 m²kg^–2.

A star 2.5 times the mass of the sun and collapsed to a size of 12 km rotates with a speed of 1.2 rev. per second. (Extremely compact stars of this kind are known as neutron stars. Certain stellar objects called pulsars belong to this category). Will an object placed on its equator remain stuck to its surface due to gravity? (Mass of the sun = 2 × 10³⁰ kg).

As you have learnt in the text, a geostationary satellite orbits the earth at a height of nearly 36,000 km from the surface of the earth. What is the potential due to earth’s gravity at the site of this satellite? (Take the potential energy at infinity to be zero). Mass of the earth = 6.0 × 10²⁴ kg, radius = 6400 km.

Two heavy spheres each of mass 100 kg 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 centers of the spheres? Is an object placed at that point in equilibrium? If so, is the equilibrium
stable or unstable?

Two stars each of one solar mass (= 2× 10³⁰ kg) are approaching each other for a head on collision. When they are a distance 109 km, their speeds are negligible. What is the speed with which they collide? The radius of each star is 104 km. Assume the stars to remain undistorted until they collide. (Use the known value of G).

A satellite orbits the earth at a height of 400 km above the surface. How much energy must be expended to rocket the satellite out of the earth’s gravitational influence? Mass of the satellite = 200 kg; mass of the earth = 6.0 ×10²⁴ kg; radius of the earth = 6.4 ×10⁶ m; G = 6.67 × 10^–11 N m² kg^–2.

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. 

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 = 6.4 × 10⁶ m; G= 6.67 × 10^–11 N m² kg^–2.

Assuming the earth to be a sphere of uniform mass density, how much would a body weigh half way down to the centre of the earth if it weighed 250 N on the surface?

A body weighs 63 N on the surface of the earth. What is the gravitational force on it due to the earth at a height equal to half the radius of the earth?

A Saturn year is 29.5 times the earth year. How far is the Saturn from the sun if the earth is 1.50 ×10⁸ km away from the sun?

How will you ‘weigh the sun’, that is estimate its mass? The mean orbital radius of the earth around the sun is 1.5 × 10⁸ km.

A rocket is fired from the earth towards the sun. At what distance from the earth’s centre is the gravitational force on the rocket zero? Mass of the sun = 2 ×10³⁰ kg, mass of the earth = 6 × 10²⁴ kg. Neglect the effect of other planets etc. (orbital radius = 1.5 × 10¹¹ m).

Choose the correct answer from among the given ones:
For the problem 8.10, the direction of the gravitational intensity at an arbitrary point P is indicated by the arrow (i) d, (ii) e, (iii) f, (iv) g.

Choose the correct answer from among the given ones:The gravitational intensity at the centre of a hemispherical shell of uniform mass density has the direction indicated by the arrow (see Fig 8.12) (i) a, (ii) b, (iii) c, (iv) O.

Which of the following symptoms is likely to afflict an astronaut in space (a) swollen feet, (b) swollen face, (c) headache, (d) orientational problem?

A comet orbits the Sun in a highly elliptical orbit. Does the comet have a constant (a) linear speed, (b) angular speed, (c) angular momentum, (d) kinetic energy, (e) potential energy, (f) total energy throughout its orbit? Neglect any mass loss of the comet when it comes very close to the Sun.

 Does the escape speed of a body from the earth depend on 
(a) the mass of the body, 
(b) the location from where it is projected, 
(c) the direction of projection, 
(d) the height of the location from where the body is launched?

 Choose the correct alternative:
(a) If the zero of potential energy is at infinity, the total energy of an orbiting satellite is negative of its kinetic/potential energy.
(b) The energy required to launch an orbiting satellite out of earth’s gravitational influence is more/less than the energy required to project a stationary object at the same height (as the satellite) out of earth’s influence.

 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.

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. Show that the mass of Jupiter is about one-thousandth that of the sun.

Suppose there existed a planet that went around the sun twice as fast as the earth.What would be its orbital size as compared to that of the earth?

 Choose the correct alternative:
(a) Acceleration due to gravity increases/decreases with increasing altitude.
(b) Acceleration due to gravity increases/decreases with increasing depth. (assume the earth to be a sphere of uniform density).
(c) Acceleration due to gravity is independent of mass of the earth/mass of the body.
(d) The formula –G Mm(1/r₂– 1/r₁) is more/less accurate than the formula mg(r₂– r₁) for the difference of potential energy between two points r₂and r₁distance away from the centre of the earth.

(a) You can shield a charge from electrical forces by 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?
(c) If you compare the gravitational force on the earth due to the sun to that due to the moon, you would find that 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?