Physics Ch8 Gravitation Notes Class 11

Gravitation is the force of attraction between any two masses in the universe. It governs the motion of planets, satellites, stars, and all objects on Earth. This chapter from cbse syllabus class 11 physics explains Newton’s universal law of gravitation, acceleration due to gravity, gravitational field and potential, motion of satellites, and Kepler’s laws of planetary motion.

Universal Law of Gravitation

Sir Isaac Newton was the first to propose that the same force which causes an apple to fall to the ground also governs the motion of celestial bodies. According to him, any two objects in the universe attract each other with a force that:

1. Is directly proportional to the product of their masses.
2. Is inversely proportional to the square of the distance between them.
3. Acts along the line joining the two bodies.

Mathematically, F = Fm₁m₂/r²; where:

• F = gravitational force
• m₁, m₂ = masses of the two objects
• r = distance between their centres
• G = universal gravitational constant

The value of F was first calculated by Cavendish in 1798 using the torsion balance experiment. Its value is ⇒ G = 6.67 x 10^-11 Nm²kg^–2.

Importance of the Universal Law of Gravitation

The law is universal as it applies to all objects in the universe, irrespective of size and distance. It explains:

• Motion of planets around the Sun
• Motion of the Moon around the Earth
• The phenomena of tides due to the Moon and Sun
• Artificial satellite orbits
  
• Falling of objects on Earth

Acceleration Due to Gravity (g)

When an object falls freely near the Earth’s surface, it experiences an acceleration directed towards the center of the Earth. This is called acceleration due to gravity, denoted by g.

For an object of mass mm on Earth:

For an object of mass m on Earth ⇒ F = GMm/R²; where M is the mass of Earth and R is the radius of Earth.

But F = mg, therefore: g = GM/R². Thus, g depends on G, M, and R. The average value of g on Earth is approximately 9.8 m/s².

Variation of g

Acceleration due to gravity is not constant everywhere. It varies with altitude, depth, latitude and shape of Earth.

(i) Variation with Altitude

At height h above Earth’s surface ⇒ g_h = GM/(R + h)² = g(R²/(R + h)²)

If h<<R: g_h ≈ g(1 – 2h/R) Thus, g decreased with altitude.

(ii) Variation with Depth

At depth d below Earth’s surface ⇒ g_d ≈ g(1 – d/R) Thus, g decreases linearly with depth and becomes zero at the center of the Earth.

(iii) Variation with Latitude

Due to Earth’s rotation, effective gravity decreases as we move from poles to equator.

At latitude Φ: g’ = g – Rω²cos²Φ; where ω is angular velocity of Earth. At the equator, reduction is maximum, and at poles, there is no effect.

Gravitational Field

A gravitational field is the region in space around a mass in which another mass experiences a force of attraction. The gravitational field strength or gravitational intensity at a point is defined as the gravitational force experienced by a unit mass placed at that point.

E = F/m = GM/r²

Gravitational Potential Energy

The gravitational potential energy of a body of mass mm at a distance rr from the center of Earth is ⇒ U = – GMm/r. The negative sign indicates that the potential energy is taken as zero at infinity and decreases as the body comes closer to Earth.

For small height h ⇒ U = mgh

Gravitational Potential

Gravitational potential at a point is the work done in bringing a unit mass from infinity to that point without acceleration.

V = – GM/r

Relation between field and potential ⇒ E = – dV/dr

Escape Velocity

The escape velocity is the minimum velocity with which a body must be projected from Earth’s surface so that it can escape Earth’s gravitational pull and never return.

Derivation:

Condition: Total energy ≥ 0 ⇒ 1/2mv_e² – GMm/R = 0 ⇒ v_e = √(2GM/R) = √2gR

For Earth, v_e ≈ 11.2km/s. It is independent of the mass of the body.

Orbital Velocity of Satellites

A satellite revolves around Earth in a circular orbit due to the balance of gravitational force and centripetal force ⇒ GMm/r² = mv²/r ⇒ v = √GM/r. This velocity is called orbital velocity. For a satellite close to Earth’s surface: v ≈ 7.9km/s.‍

Time Period of Satellite

The orbital period is given by ⇒ T = 2πr/v = 2π√r³/GM. This is also called Kepler’s Third Law of Planetary Motion.‍

Height of Geostationary Satellites

A geostationary satellite appears stationary relative to Earth. It has:

• Time period = 24 hours
• Rotates in equatorial plane
• Same angular velocity as Earth

From T = 2π√r³/GM, the height comes out to be about 36,000 km above Earth’s surface.

Energy of Satellite

The total energy of a satellite in orbit is the sum of its kinetic and potential energies.

K = 1/2 mv² = GMm/2r; and U = – GMm/r ⇒ E = K + U = – GMm/2r

Thus, energy is negative, indicating a bound system.

Kepler’s Laws of Planetary Motion

Johannes Kepler proposed three laws:

1. Law of Orbits - Planets revolve around the Sun in elliptical orbits with the Sun at one focus.
2. Law of Areas - A line joining the planet and the Sun sweeps out equal areas in equal intervals of time.
3. Law of Periods - Square of the orbital period is directly proportional to the cube of the semi-major axis. Consequently, T² ∝ r³.

Newton later proved these laws mathematically using his law of gravitation.

Weightlessness

Weight is the force with which a body is attracted towards the Earth. In an orbiting satellite, both the satellite and the body inside it fall freely under gravity. Hence, relative to each other, they appear weightless. This condition is called weightlessness.

Conclusion

Gravitation explains motion from falling objects to planetary orbits. Understanding Newton’s law, variation of g, gravitational potential, and satellite motion is essential for solving numerical problems and understanding celestial mechanics.

FAQs

Q1. What is escape velocity?

Ans. It is the minimum velocity required for a body to escape from Earth’s gravitational field without any further acceleration. For Earth, it is about 11.2 km/s.

Q2. What are geostationary satellites?

Ans. Satellites that revolve around the Earth in 24 hours in the equatorial plane in the same direction as Earth’s rotation. They appear stationary with respect to Earth.

Q3. What is the difference between gravitational force and electrostatic force?

Ans. Gravitational force is always attractive, very weak, and acts between masses. Electrostatic force can be attractive or repulsive, much stronger, and acts between charges.

Q4. Does the value of g remain constant everywhere on Earth?

Ans. No, the value of g varies with altitude, depth and latitude of the place.

Q5. What is the difference between mass and weight?

Ans. Mass is the amount of matter in a body, constant everywhere, scalar. Weight is the force with which Earth attracts a body, depending on g, vector.

Q6. What is the difference between orbital velocity and escape velocity?

Ans. Orbital velocity keeps the body in circular orbit. Escape velocity removes the body completely from the gravitational field.