Gravitation Class 11 Physics Notes | Chapter 8 Revision

Key Definitions in Gravitation

Gravitational Force

The attractive force existing between any two objects possessing mass, acting along the line joining their centers.

Universal Gravitational Constant (G)

A fundamental constant of proportionality in Newton's Law of Gravitation, numerically equal to 6.67 × 10⁻¹¹ Nm²/kg².

Acceleration Due to Gravity (g)

The acceleration experienced by an object falling freely under the influence of a celestial body's gravitational pull, typically Earth's (approx. 9.8 m/s² on Earth's surface).

Gravitational Field Intensity

The gravitational force experienced per unit mass at a particular point within a gravitational field. It is a vector quantity.

Gravitational Potential Energy (U)

The energy possessed by a body due to its position in a gravitational field, defined as the work done by an external agent to bring the body from infinity to that position.

Gravitational Potential (V)

The gravitational potential energy per unit mass at a point in a gravitational field. It is a scalar quantity.

Escape Velocity (v_e)

The minimum velocity an object must attain to completely escape the gravitational pull of a celestial body and move infinitely far away, never returning.

Orbital Velocity (v_o)

The specific horizontal velocity required for an object (satellite) to maintain a stable circular orbit around a celestial body without falling or escaping.

Newton's Universal Law of Gravitation

Newton's Universal Law of Gravitation states that every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This fundamental law is expressed mathematically as:

**F = G (m₁ m₂) / r²**

Where:

• F is the gravitational force between the two masses.
• G is the Universal Gravitational Constant (6.67 × 10⁻¹¹ Nm²/kg²).
• m₁ and m₂ are the masses of the two interacting objects.
• r is the distance between their centers.

Key characteristics of gravitational force:

• It is always attractive.
• It acts along the line joining the centers of the two masses.
• Gravitational forces form an action-reaction pair (Newton's Third Law).
• It is a conservative force, meaning the work done by it is independent of the path taken.
• It is a long-range force, although its strength diminishes rapidly with distance.
• It is the weakest of the four fundamental forces but is dominant on a cosmic scale due to the enormous masses of celestial bodies. The value of 'G' is independent of the medium between the masses.

Acceleration Due to Gravity (g) and its Variations

The acceleration due to gravity (g) is the acceleration produced in a freely falling body due to the gravitational pull of a celestial body, typically Earth. Near the Earth's surface, its average value is approximately 9.8 m/s². This value is not constant and varies significantly across the Earth and at different altitudes and depths.

The fundamental formula for 'g' on the surface of a spherical body of mass M and radius R is given by: g = GM/R², where G is the universal gravitational constant.

Several factors cause variations in 'g':

1. Variation with Altitude (height h above surface): As an object moves further away from the Earth's surface, its distance 'r' from the center of Earth increases (r = R+h). Consequently, 'g' decreases with altitude. The general formula is **g' = g [R / (R+h)]². For small heights (h << R), a simplified approximation is g' ≈ g (1 - 2h/R)**.
2. Variation with Depth (depth d below surface): As an object moves inside the Earth, the mass of the Earth effectively attracting it decreases (only the mass of the sphere of radius R-d attracts it). This also causes 'g' to decrease. The formula is **g' = g * (1 - d/R). At the center of the Earth (d=R), 'g' becomes zero**.
3. Variation with Latitude (λ): Due to the Earth's rotation, a part of the gravitational force provides the necessary centripetal force for an object to move in a circle with the Earth. This causes a reduction in the effective 'g'. The formula for effective acceleration due to gravity is g' = g - Rω²cos²λ, where ω is the angular velocity of Earth and λ is the latitude. This means 'g' is maximum at the poles (λ = 90°, cosλ = 0) and minimum at the equator (λ = 0°, cosλ = 1).
4. Variation due to Shape of Earth: The Earth is not a perfect sphere; it is an oblate spheroid, bulging at the equator and flattened at the poles. This means the equatorial radius (R_e) is greater than the polar radius (R_p). Since g ∝ 1/R², acceleration due to gravity is less at the equator and more at the poles.

Gravitational Potential Energy and Potential

Kepler's Laws of Planetary Motion

• First Law (Law of Orbits): All planets move in elliptical orbits with the Sun located at one of the foci.
• Second Law (Law of Areas): The line joining a planet to the Sun sweeps out equal areas in equal intervals of time. This implies that a planet moves faster when it is closer to the Sun and slower when it is farther away (conservation of angular momentum).
• Third Law (Law of Periods): The square of the orbital period (T) of any planet is directly proportional to the cube of the semi-major axis (a) of its elliptical orbit. * T² ∝ a³ or **T² = (4π²/GM) * a³** (for orbit around a mass M)

Satellite Motion and Escape Velocity

Key Points to Remember for Gravitation

• Gravitational force is always attractive, conservative, and acts along the line joining the centers of masses.
• The Universal Gravitational Constant 'G' is a scalar and its value (6.67 × 10⁻¹¹ Nm²/kg²) is constant everywhere, independent of medium or temperature.
• 'g' (acceleration due to gravity) is a vector quantity and varies with altitude, depth, latitude, and the non-spherical shape of Earth.
• At the center of the Earth, the acceleration due to gravity 'g' is zero.
• Gravitational potential and potential energy are always negative (or zero at infinity) for an attractive force.
• Kepler's Second Law (Law of Areas) is a direct consequence of the conservation of angular momentum.
• The total mechanical energy of a satellite in a stable orbit is always negative, indicating it is bound to the central body.
• Escape velocity is independent of the mass or direction of projection of the projectile.
• Geostationary satellites have an orbital period of 24 hours, appearing stationary relative to a point on Earth's surface.

Worked Example

• {"title":"Gravitational Force Calculation","bodyMarkdown":"Q: Two bodies of masses 10 kg and 100 kg are separated by a distance of 1 m. Calculate the gravitational force between them. (Given G = 6.67 × 10⁻¹¹ Nm²/kg²)\n\nA:\nGiven: m₁ = 10 kg, m₂ = 100 kg, r = 1 m, G = 6.67 × 10⁻¹¹ Nm²/kg²\nUsing Newton's Law of Gravitation: F = G (m₁ m₂) / r²\nF = (6.67 × 10⁻¹¹) (10 100) / (1)²\nF = (6.67 × 10⁻¹¹) * 1000\nF = 6.67 × 10⁻⁸ N\n\nThe gravitational force between the two bodies is 6.67 × 10⁻⁸ N."}

Exam Tip: Avoiding Common Traps

When solving numerical problems involving variation of 'g', be careful to use the correct formula for altitude (h) versus depth (d). Remember that for altitudes, the general formula g' = g [R / (R+h)]² is always applicable, while g' = g (1 - 2h/R) is an approximation for h << R. For depth, g' = g * (1 - d/R). Always check the units and convert them to SI units (meters, kilograms, seconds) before calculation. Pay attention to whether the question asks for gravitational potential (scalar) or gravitational field intensity (vector).

Practice Questions with Solutions

• Is the gravitational force between two bodies affected by the nature of the medium separating them? No, gravitational force is independent of the intervening medium.
• What happens to the acceleration due to gravity 'g' at the center of the Earth? At the center of the Earth, 'g' becomes zero.
• State the principle behind Kepler's Second Law of Planetary Motion. Kepler's Second Law (Law of Areas) is a direct consequence of the conservation of angular momentum.
• What is the relationship between escape velocity and orbital velocity for an object launched from the surface of a planet? Escape velocity (v_e) is √2 times the orbital velocity (v_o) for an object orbiting very close to the planet's surface (v_e = √2 * v_o).

Frequently Asked Questions