## Kerala Plus One Physics Notes Chapter 8 Gravitation

Introduction:

Gravitational force is one of the fundamental forces that exist in nature. It is the force that holds you to the Earth, holds the Moon in orbit around the Earth, and holds Earth in orbit around the Sun. it is responsible for holding billions and billions of stars in the Galaxy and the countlesss molecules and dust particles between stars.

Kepler’s Laws

First law (law of orbits):

All planets move around the sun in elliptical orbits with the sun at one of the foci of that ellipse.

Second law (law of areas):

The line joining the planet and the sun sweeps equal areas in equal intervals of time. Thus, planets move faster when they are closer to sun.

Third law (law of periods):

The square of the time period of revolution of a planet is proportional to the cube of the semi-major axis of the ellipse.

T^{2} α r^{3 }If r_{1} and r_{2} are the shortest and the longest distance of the planet from the sun, the semi-major axis is given by

where G is called the universal gravitational constant and is defined as the force of attraction between two bodies of unit mass (1 kg) each and placed unit distance (1 m) apart.

Universal Law of Gravitation:

On the basis of Kepler’s laws and observations of motion of planets, Newton gave a universal law of force acting between any two particles of matter.

Statement:

“Every body in the universe attracts each other with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.”

Explanation:

Consider two masses m_{1} and m_{2} separated by a distance r. The force of attraction between the two particles can be written as

Note:

The gravitational force between two masses is independent of the medium surrounding it.The gravitational forces between two bodies are equal and opposite. ie„ a falling apple pulls the earth towards wit’: the same force (Mg) as with which the Earth pulls it down, but as the mass of earth is huge the acceleration (a = F/M) produced is negligible which is why we don’t see the earth moving.

Gravitation is the force of attraction between any two bodies in the universe while gravity is the force of attraction between the earth and any object lying on or near its surface.

where G is called the universal gravitational constant and is defined as the force of attraction between two bodies of unit mass (1 kg) each and placed unit distance (1 m) apart.

Dimensions of

CAQ

Question 1.

Is it possible to shield a body from gravitational effects?

Answer:

No, it is not possible to shield a body from gravitational effects because gravitational interaction does not depend upon the nature of the intervening medium

Acceleration due to Gravity :

The acceleration produced in a freely falling body under the gravitational pull of the earth is called acceleration due to gravity. If the force experienced by the particle of mass

where M = mass of earth, R = radius of earth

Newton’s second law tells us that,

force = mass of body × acceleration = mg

Comparing the above equations we get,

where g is the acceleration due to gravity.

Note:

Typically near the surface of earth g = 9.8 ms^{-2}

Weight of a body

Weight of a body is defined as the gravitational force with which a body is attracted towards the centre of the earth. It is given by

As the value of g varies from place to place, the weight of a body also varies from place to place.

CAQ

Question 2.

A body weighs 90 kg f on the surface of the earth. How much will it weight on mass if its mass is 1/9 and the radius is 1/2 of that of the earth?

Answer:

The acceleration due to gravity on the surface of the earth is given by

The acceleration due to gravity on the surface of Mars is given by

Mass of the earth :

Mean density of earth:

1. Effect of altitude

If an object is placed at a distance h above the surface of the earth, the force of gravitation on it due to the earth is

The above equation shows that value of acceleration due to gravity decreases with the increase in height h. This explains why the value of g is less at mountains than at plains.

Loss in weight at height (h<<R_{3}):

2. Effect of depth

A body at point B in the figure experiences gravitational force due to the inner shaded sphere of radius (R-d) and mass M where

where g_{0} is the value of g at the surface of the earth.

Therefore the acceleration due to gravity decreases with increase in depth. This explains why acceleration due to gravity is less in mines than on earth’s surface.

Note: At the centre of earth, ie., d = R

Weight of a body with mass m at the centre of earth = m ×g = m ×o = 0

But remember that the mass remains the same.

Loss in weight at depth (d):

Using the above equation we get,

Loss in weight =

Note: The value of g decreases as we go above the surface of earth or below the surface of earth, ie.. value of g is maximum on the surface of earth. Also the value of g at a height n is same as the value of g at a depth d = 2h.

3. Effect due to rotation of earth

The acceleration due to gravity

(i) decreases due to rotation of earth

(ii) increases with the increase in lattitude. It is given by ;

where θ = latitude of the radius

So as we move from equator to pole, the acceleration due to gravity increases.

Special cases:

- At the equator,θ= 0°, cos

θ=1, g_{e}=g-Rω^{2} - At the poles, θ= 90°, cos θ =0,

g_{e}= g-Rω^{2}× 0=g

Acceleration due to gravity is minimum at the equator and maximum at the poles.

4.Effect due to shape:

The equatorial radius of earth is longer than its polar radius. The value of g increases from equator to pole. It is given as g_{pole}>g_{equator. CAQ }Question 3.

If the change in the value of g at a height h above the surface of the earth Is the same as at a depth x below it, then (both x and h being much smaller than the radius of the earth). Then obtain the relation between t and h.

Answer:

The value of g at the height h from the surface of earth

The value of g at depth x below the surface of earth

These two are given equal, hence On solving, we get x = 2h

Question 4.

When dropped from the same height a body reaches the ground quicker at poles than at the equator. Why?

Answer:

The acceleration due to gravity is more at the poles than at the equator. When the initial velocities and distances travelled are the same, the time taken by the body is smaller if the acceleration due to gravity is large. Hence, when dropped from the same height, a body reaches the ground quicker at the poles than at the equator.

Gravitational field of earth:

It is the space around the earth where its gravitational influence is felt.

Intensity of gravitational field at a point

It is defined as the force experienced by a body of unit mass (1 kg) placed at that point.

Intensity of gravitational field of a body kept at point Pat distance r from centre of earth is

Gravitational Potential Energy:

The work done in carrying a mass ‘m’ from infinity to a point at distance r is called gravitational potential energy.

The gravitational potential energy of the system is given by

Note:

The negative sign in the above equation indicates that gravitational force between the earth and the body is attractive in nature.

This is because as work is done the energies of both bodies are spent, so their energy decreases by this amount.

Gravitational Potential:

Gravitational potential at a point is defined as the amount of work done in bringing a body of unit mass (1 kg) from infinity to that point.

Gravitational potential,

SI unit is Jkg^{-1}.

Dimensional formula is [M°L^{2}T^{2}].

Gravitational potential at a point due to the earth:

The work done in bringing a body of mass m from infinity to a point at distance r from the centre of earth is

Therefore the gravitational potential due to the earth at distance r from its centre is

At the surface of earth ,r=R

Note:

Gravitational potential energy = Gravitational potential x mass

CAQ

Question 5.

The gravitational field intensity at a point 10,000 km from the centre of the earth is 4.8 N kg^{-1}. Calculate the gravitational potential at that point.

Answer:

Escape Velocity:

It is the minimum velocity with which a body must be projected vertically upwards to just escape the gravitational field of the earth.

Consider a body of mass m, leaving the surface of earth with escape velocity v. This projectile body has a kinetic energy K=1/2 mv^{2} and a potential energy given by,

in which M is the mass of the earth and R its radius.

When this body reaches inifinity, its total energy will be zero. This is because at infinity, the body stops and thus kinetic energy becomes zero. As there is infinite separation between the two bodies the two bodies are our zero-potential energy configuration.

Note:

The escape velocity does not depend upon the mass of the projectile body.

Escape velocity of earth:

Note:

Here we haven’t considered air resistance, in real practice the escape velocity will be a little greater than the above value.

Satellites:

Satellites revolve around planets. A number of rockets are fired from the earth to

establish the satellite in the desired orbit . Once the satellite is placed in the desired orbit with correct speed for that orbit, it will continue to move in that orbit under the gravitational attraction of the earth.

Orbital Velocity:

It is the velocity required to put the satellite into its orbit around the earth.

The gravitational force on the satellite is

The centripetal force required by the satellite to stay in this orbit is

In equilibrium the centripetal force is given by the gravitational force

This orbit in which the satellite revolves is called minimum orbit. The velocity corresponding to minimum orbit is called first cosmic velocity.

The value of first cosmic velocity is 7.92 ; km/s.

Note:

The escape velocity is related to orbital velocity as follows v_{e} = √2 v_{0}; where v_{e} is the escape velocity of the body.

Time Period of a Satellite:

It is the time taken by the satellite to revolve once around the earth. It is given by

Energy of an Orbiting Satellite:

The kinetic energy of the satellite in a circular orbit at height h from surface of earth with speed v is

The potencial energy at distance (R+h) from

Note:

The negative sign means that the satellite is bound to the earth Is at infinity the total energy is thus we have to give energy to satellite to send it to infinity to make total energy zero This shows that the satellite is bound to the earth.

CAQ

Question 6.

An artificial satellite revolves around the earth at a height of 1000 km. The radius of the earth Is 6.38 x 10^{3} km. Mass of the earth is 6 × 10^{24 }kg and G = 6.67 × 10^{-11} . Find Its orbital velocity and period of revolution.

Answer:

Geostationary Satellite:

The satellite having the same time period of revolution as that of the earth is called geostationary satellite. As seen from earth, this satellite will appear to be stationary. Such satellites should rotate in the equatorial plane from west to east. The orbit of a geostationary satellite is called parking orbit. These satellites are used for communication and weather for- casting.

Weightlessness :

When there is no earth or other heavy body to attract us, we feel weightless. Objects feel weightless in space. When we fall freely under the earth’s gravity, then also we feel weightless. Let us try this, we stand in a lift. One of us stand on a weighing meter. We read the weight. When the lift moves down, we notice the weight decreasing. Now, if ; for a few seconds the lift falls down freely (may this not happen to you) we see that the weighing meter reads zero.

eg., Consider an astronaut (or space-man) of mass m is present in the artificial satellite. When the satellite is orbiting around earth, the man in the satellite experience a centrifugal force whose direction is away from the centre. Therefore the two forces acting are

Hence an astronaut feels weightlessness in an artificial satellite

Did you know? A Black hole is a region of space having a gravitational field so intense that no matter or radiation such as light can escape. These are formed when massive stars collapse into very negligible volumes (thus radii) making its escape velocity equal to or greater than the speed of light (max. speed possible) making it impossible for matter or radiation to escape its feild.

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