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HSSLIVE Plus One Physics Chapter 8: Gravitation Notes

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Gravitation explores one of the fundamental forces of nature, from Kepler’s empirical laws of planetary motion to Newton’s universal law of gravitation. Students discover how gravitational interactions govern everything from falling objects on Earth to the motion of planets and satellites. This chapter connects gravitational potential energy to orbital mechanics and escape velocity, providing insights into space exploration, satellite technology, and the structure of our solar system.

Plus One Physics Notes – Chapter 8: Gravitation

SCERT Kerala Board

Universal Law of Gravitation

Newton’s Law of Gravitation: Every object in the universe attracts every other object with a force that is:

  • Directly proportional to the product of their masses
  • Inversely proportional to the square of the distance between them

Mathematical expression: F = G (m₁m₂)/r²

  • F = gravitational force (N)
  • G = universal gravitational constant (6.67 × 10⁻¹¹ N·m²/kg²)
  • m₁, m₂ = masses of the two objects (kg)
  • r = distance between the centers of the objects (m)

Characteristics:

  • It is a central force (acts along the line joining the centers of masses)
  • It is always attractive
  • It is a long-range force
  • It is the weakest of the four fundamental forces

Acceleration Due to Gravity

Definition: The acceleration experienced by an object due to the gravitational force of a large body like Earth.

Value at Earth’s surface: g = 9.8 m/s²

Variation with height: g_h = g(R/(R+h))²

  • g_h = acceleration due to gravity at height h
  • g = acceleration due to gravity at surface (9.8 m/s²)
  • R = radius of Earth (6.37 × 10⁶ m)
  • h = height above Earth’s surface

Variation with depth: g_d = g(1-d/R)

  • g_d = acceleration due to gravity at depth d
  • d = depth below Earth’s surface

Gravitational Potential Energy

Definition: The energy possessed by an object due to its position in a gravitational field.

Formula (near Earth’s surface): PE = mgh

  • m = mass of the object (kg)
  • g = acceleration due to gravity (9.8 m/s²)
  • h = height above the surface (m)

General formula: PE = -G(m₁m₂)/r

  • G = universal gravitational constant
  • m₁, m₂ = masses
  • r = distance between centers

Escape Velocity

Definition: The minimum velocity needed for an object to escape the gravitational pull of a celestial body.

Formula: v_e = √(2GM/R)

  • v_e = escape velocity (m/s)
  • G = universal gravitational constant
  • M = mass of the celestial body (kg)
  • R = radius of the celestial body (m)

For Earth: v_e ≈ 11.2 km/s

Orbital Velocity and Period

Orbital velocity: v = √(GM/r)

  • v = orbital velocity (m/s)
  • G = universal gravitational constant
  • M = mass of the central body (kg)
  • r = orbital radius (m)

Time period: T = 2π√(r³/GM)

  • T = time period (s)
  • r = orbital radius (m)
  • G = universal gravitational constant
  • M = mass of the central body (kg)

Kepler’s Laws of Planetary Motion

  1. Law of Orbits: All planets move in elliptical orbits with the Sun at one focus.
  2. Law of Areas: A line joining a planet to the Sun sweeps out equal areas in equal times.
  • Consequence: Planets move faster when closer to the Sun and slower when farther away
  1. Law of Periods: The square of the time period of a planet is directly proportional to the cube of the semi-major axis of its orbit.
  • T² ∝ r³
  • For circular orbits: T² = (4π²/GM)r³

Gravitational Potential

Definition: Gravitational potential at a point is the work done per unit mass in bringing a small test mass from infinity to that point.

Formula: V = -GM/r

  • V = gravitational potential (J/kg)
  • G = universal gravitational constant
  • M = mass of the attracting body (kg)
  • r = distance from the center of the attracting body (m)

Satellite Motion

Types of satellites:

  • Geostationary satellite: Orbits above the equator at a height of 36,000 km with a period of 24 hours
  • Polar satellite: Passes over the poles in a north-south orbit

Energy of a satellite: Total Energy E = Kinetic Energy + Potential Energy = -GMm/(2r)

Weight and Weightlessness

Weight: The gravitational force exerted by a celestial body on an object

  • W = mg

Weightlessness: Condition when apparent weight becomes zero

  • Occurs in free fall
  • Experienced by astronauts in orbiting spacecraft

Practice Problems

  1. Calculate the gravitational force between two objects with masses 50 kg and 70 kg placed 2 meters apart.
  2. Find the escape velocity on a planet with twice the mass and half the radius of Earth.
  3. Calculate the orbital velocity of a satellite orbiting at an altitude of 1000 km above Earth’s surface (Earth’s radius = 6370 km).

These notes cover the essential concepts of Gravitation for Plus One Physics following the SCERT Kerala Board syllabus, presented in a simplified manner for easier understanding.

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