Physics Class 12 Gravitation - Kepler's Laws, Universal Law of Gravitation, Escape Velocity

Gravitation - Physics can be a very tough subject for all the board examination students so every unit which is included in the physics subjects must be taken care of. The students can start their preparation for the physics subjects by taking into consideration the unit of Gravitation. Check out the important topics included in the Gravitation unit for physics subject from the article provided below and prepare yourself accordingly from the given study notes:

Gravitation - Kepler's Laws Of Planetary Motion

The German astronomer Johannes Kepler presented Kepler's law of planetary motion which describes the motion of the planets in the solar system. The three laws of planetary motion are as follows:

• All planets move about the sun in elliptical orbits having the sun as one of the focus.
• A radius vector joining any planet to the sun sweeps out equal areas in equal lengths of time.
• The squares of the sidereal periods (of revolution) of the planets are directly proportional to the cubes of their mean distance from the Sun.

• Physics Class 12 Units and Measurements
• Physics Class 12 Motion in a Straight Line
• Physics Class 12 Laws of Motion

Universal Law of Gravitation 

Sir Isaac Newton presented the universal law of gravitation in 1687 and it is used to explain the motion of the planets and the moon. The universal law of gravitation states that every particle attracts every other particle in the universe with force directly proportional to the product of the masses and inversely proportional to the square of the distance between them. 

Formula For Universal Law of Gravitation

The formula for the universal law of gravitation is f equals G into m1m2/r2

Where F is the gravitational force between bodies, m1 and m2 are the masses of the bodies, r is the distance between the centers of two bodies, and G is the universal gravitational constant.

The constant proportionality (G) in the above equation is known as the universal gravitation constant. Henry Cavendish experimentally determined the precise value of G. The value of G is found to be G = 6.673 x 10-11 N m2/kg2.

Acceleration Due To Gravity 

The gravitational acceleration depends upon the distribution of mass within Earth or any celestial body. The distribution of matter affects the geometry of the surface where the potential is constant. Gravity is the universal force of attraction that is always present between all things and the matters in the universe. Gravity is also measured by the acceleration or the movement that is occurring in a freely falling object. The acceleration of gravity at the Earth’s surface is approximately 9.8 m/s2. As a result, for every second that an item is in free fall, its speed rises by approximately 9.8 m/s2. 

Formula For Acceleration Due To Gravity

We know that the force on anybody is given by,

F = mg

where F is the force acting, g is the acceleration due to gravity, and m is the mass of the body.

And according to the universal law of gravitation, 

F = GMm/(r+h)2

where F is the force between two bodies, G is the universal gravitational constant, m is the mass of the object, M is the mass of the earth, r is the radius of the earth, and h is the height above the surface of the earth.

Since the height is negligibly small compared to the radius of the earth, rearrange the above expression as,

F = GMm / r2

Now equating both expressions,

mg = GMm / r2

⇒ g = GM / r2

Acceleration Due To Gravity Variation With Altitude

Consider a sample mass (m) at a height (h) above the earth’s surface. Now, the gravitational force exerted on the test mass is:

F = GMm / (R+h)2

Where R and M are the radii and the mass of the earth. Then, the acceleration due to gravity at a certain height is ‘h’. So, mgh = GMm / (R+h)2

⇒ gh = GM / [R2 (1+ h/R)2] ⇢ (1)

Now, the value of g is,

g = GM/R2 ⇢ (2)

On dividing equations (2) and (1), we get,

gh = g (1+h/R)-2 ⇢ (3)

This is the acceleration due to gravity at a height above the earth’s surface. According to the above formula, the value of g decreases with the increasing height of an object and becomes zero at an infinite distance from the earth.

Acceleration Due To Gravity Variation With Depth

Suppose we consider a body of mass m is at a point B where B is at a depth of h from the earth’s surface, and its distance from the center is R – h.

Now, we have, 

gd = g (R – d)/R

The acceleration due to gravity (gd), at this depth, is given by,

h < R

or 

(1 – h/R) < 1

Therefore, gd < g

And this acceleration, (gd), decreases as we move towards the center of the earth, which is experienced only when we move very deep towards the center of the earth. Hence, the value of g changes with height and depth. But the value of g changes even on the surface of the earth also. For example, g is the highest on the poles and lowest on the equator.

Gravitational Potential Energy And Gravitational Potential

Gravitational potential energy is the energy that is possessed by an object due to the change and its position whenever it is inside a gravitational field. The amount of work done in moving a unit test mass from infinity into the gravitational influence of source mass is known as gravitational potential. 

Gravitation- Escape Velocity

Escape velocity can be defined as the speed at which the object travels far from the entire planet’s or the moon's gravity and then leaves the premises without any development of propulsion. 

Gravitation- Formula For Escape Velocity

The formula for escape velocity is √2GM/r

Where VC is the escape velocity

G is the universal gravitational constant

M is the mass of the celestial object whose gravitational pull has to be superseded

r is the distance from the object to the center of mass of the body to be escaped

Gravitation- Orbital Velocity Of A Satellite 

Orbital velocity is the speed that is needed to achieve a balance between the gravitational pull on the satellite and the inertia of a satellite's motion. The velocity of the satellite depends upon the altitude above Earth when you are near the planet, the orbital velocity is faster. For 124 miles (200 kilometers) of altitude, the required orbital velocity is a little more than 17,000 mph (about 27,400 kph). 

The important sections related to the Gravitation unit are discussed below so that the students can start preparing for the board examination. Make sure that you are covering each and every important topic for the physics subjects before diving into your board exams.