Gravity: Definition, Effect, Acceleration due to Gravity

WHAT IS GRAVITY

Gravity is a force that pulls two objects towards each other. Everything that has mass also has gravity. The more massive the object, the stronger its attraction. Earth‘s gravity keeps you on the ground and causes objects to fall. Gravity is what keeps the planets around the sun and the moon around the earth. The closer you are to an object, the stronger its attraction. Gravity gives you weight. This is the force that pulls the entire mass of your body together.

Definition: The force with which the earth attracts any object on or near the surface is called gravity.

As a result of this gravity, when an object is released from some point above the earth’s surface, the object moves towards the earth’s surface. Obviously, gravity is just a special form of gravitation

WHAT IS GRAVITY

Let us assume mass of Earth = M, radius = R, a body has a mass m lies at a distance r from the center O of the earth. Assuming the Earth to be a homogeneous sphere of radius R, all the mass M of the Earth can be assumed to be concentrated at its center. Then according to the law of gravitation, the force of attraction of the earth on the object, i.e. the amount of gravitational force,

F = GMm/r² …….(I)

According to Newton’s second law of motion, when a force F is applied to an object of mass m, an acceleration of the object is created in the direction of F i.e. towards the center of the earth. This acceleration due to the force of gravity is the gravitational acceleration. It is denoted by g.

Definition: The acceleration due to the force of gravity on a freely falling object is called gravitational acceleration.

For an object of mass m, we get from the equation F = ma, F = mg Hence, comparing equation (I) no,

mg = GMm/r² or, g = GM/r² …….(II)

Since and are both constants,

g ∝ 1/r² ………..(III)

That is, the value of the gravitational acceleration at a point near the Earth‘s surface is directly proportional to the square of the distance from the center of the Earth.

(II) It is clear from equation no that the value of gravitational acceleration at any point does not depend on the mass (m of) of the object. All objects, light or heavy, have the same value of gravitational acceleration at a given location. That is, the value of g is the same at places equidistant from the center of the earth.

r = R for any point above the Earth’s surface. Hence from equation (II) the surface value of gravitational acceleration is obtained,

g = GM/R² ……(IV)

This is the relationship between the gravitational acceleration g and the gravitational constant G.

If the average density of the Earth is ρ, M = 4/3πR³ρ

So, g = (G·4/3πR³ρ)/R² or, ρ = 3g/4πRG ………(V)

This is the relationship between the gravitational constant G and the average density ρ of the Earth.

The value of gravitational acceleration at the surface can be determined from equations (IV) and (V) or with the help of other experiments.

CGS — g = 980.6 cm/s²

FPS — g = 32.2 ft/s²

SI — g = 9.806 m/s²

VARIATION IN ACCELERATION DUE TO GRAVITY

Gravitational acceleration (g) is not a constant. The reasons for this are,

For the surface to be ellipsoidal, the value of g varies at different places on the surface.
The value of g varies at different heights above the ground.
The value of g varies at different depths from the surface.

EFFECT OF EARTH'S ELLIPSOIDAL SURFACE

The surface is not completely spherical, but ellipsoidal. The north and south polar regions are slightly depressed and the equator is slightly inflated. As a result, the Earth’s equatorial radius is slightly greater than its polar radius (about 21 km). We have seen, g ∝ 1/r² i.e. the value of g is directly proportional to the square of the distance from the centre of the earth to a point on the surface. Thus, as the distance from the Earth’s centre to the poles is less, the value of g is greater there; On the other hand, as the distance to the equatorial region is greater, the value of g is lower there.

VARIATION WITH ALTITUDE

Let the value of gravitational acceleration of the surface = g; Gravitational acceleration at a point P at a height h above the surface = g’ and the distance from the centre of the earth O to the surface, i.e. the radius of the earth = R. Therefore, distance from O to P = R+h. Since the value of gravitational acceleration at a place is directly proportional to the square of the distance from the center of the earth, g’/g = R²/(R+h)² ………(I)

or, g’/g = 1/[1+(h/R)]² = [1+(h/R)]-2 = (1- 2h/R)

So, g’ = g(1- 2h/R) ……..(II)

This is the relationship between the gravitational acceleration g’ at a height h above the surface and the gravitational acceleration g at the surface.

VARIATION WITH DEPTH

Let us assume that an object of mass m is placed at a point P of radius R and depth h from the earth’s surface. The point O is the center of the earth. A sphere of radius (R-h) is imagined centered at point O. This imaginary spherical surface divides the Earth into two parts: (i) an inner sphere of radius (R-h) and (ii) a spherical shell of thickness h. In this case, it can be proved that an object of mass m is located just outside the inner sphere and experiences a gravitational force for the sphere, but does not experience a gravitational force for the sphere because it is located inside the spherical shell. That is, in determining the gravitational force, only the inner sphere has to be calculated.

Average density of Earth, ρ = mass of Earth / volume of Earth = M/(4/3πR³) = 3M/4πR³

Volume of inner sphere = 4/3π(R-h)³

So, the mass of inner sphere, M’ = 4/3π(R-h)³ρ = 4/3π(R-h)³·3M/4πR³ = M·(R-h)³/R³

Hence the measure of the gravitational force acting on an object of mass m is,

F = GM’m/(R-h)² = GMm(R-h)/R³

Hence if the value of gravitational acceleration at point P, i.e. at depth h, is g’,

g’ = F/m = GM(R-h)/R³ = GM/R²·(R-h)/R = g(R-h)/R = g[1-(h/R)] ……….(III)

Here g = GM/R² = value of gravitational acceleration at the surface.

ACCELERATION OF OBJECT AND ACCELERATION OF EARTH

Let us assume that the earth of mass M attracts an object of mass m towards itself with a force F. According to the law of gravitation, the object also attracts the earth towards itself with the help of the same F force. (This is an example of action and reaction being equal and opposite).

So according to Newton’s second law of motion,

Acceleration of Object = Force applied on Object / Mass of Object = F/m

Again, Acceleration of Earth = Force applied on Earth / Mass of Earth = F/M

So, Acceleration of Object / Acceleration of Earth = (F/m) / (F/M) = M/m

Each of the common objects near the Earth’s surface has a mass that is negligible compared to the mass of the Earth,

That is, M/m >> 1

So, Acceleration of Object / Acceleration of Earth >> 1 or, Acceleration of Object >> Acceleration of Earth

That is, since the acceleration of the object is many times greater than the acceleration of the Earth, the object itself is actually moving towards the Earth; The motion of the Earth towards the object is so negligible that it is never felt.

CONCLUSION

A gravitational force is a force of attraction between two or more things. Gravitational force increases as an object’s mass grows. Gravitational force reduces as the distance between the items increases. Gravitational force holds everything in place on Earth.

REMINDER RELATED THIS TOPIC

The force by which any two particles of matter in this universe attract each other is called gravitation.
Any two particles in the universe attract each other along their connecting straight lines. The value of this attractive force is proportional to the product of the masses of the two particles and inversely proportional to the square of the distance.
The force with which two particles of unit mass attract each other at a distance of unit distance is called the gravitational constant.
The gravitational force acting on an object of unit mass at any point in a gravitational field is called the gravitational field strength at that point.
The amount of work done by an external agent to bring an object of unit mass from an infinite distance to a point in the gravitational field is called the gravitational field at that point.
The force with which the earth attracts any object on or near the surface is called gravity. The acceleration due to the force of gravity on a freely falling object is called gravitational acceleration.
Kepler’s Laws of Motion of Planets and Satellites:

First Law: Each planet revolves around the Sun in an elliptical orbit and the Sun is at one focus of that ellipse.
Second Law: A straight line joining the Sun and a planet traverses equal areas in equal periods of time.
Third Law: The square of the orbital period of a planet around the Sun is proportional to the cube of the semi-radius of the orbit.

The minimum velocity at which an object can be thrown from the surface of the Earth or any other planet or satellite beyond its gravitational attraction is called the release velocity.
If the relative angular velocity of an artificial satellite with respect to the Earth’s angular momentum is zero and the satellite is always at the equator, it appears from the surface that the satellite is fixed in the same place in the sky. Such satellites are called geosynchronous satellites.

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22 June 2024 at 17:35

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