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Gravitation

Ch. 5Std 11

Easy Overview

Why does the Moon orbit the Earth instead of flying off into space? Why do all objects fall at the same rate regardless of their mass? And how do we know the mass of the Sun without ever going there? The answer to all these questions is gravity - the invisible force that pulls everything with mass toward everything else. Newton's law of universal gravitation was a revolution: it unified the physics of falling apples on Earth with the motion of planets in the sky. For the first time, humans understood that the same force that makes things fall also holds the solar system together. This chapter takes you from Kepler's observations of planetary motion to Newton's gravity equation, from gravitational potential energy to escape velocity, and from Earth's gravity to satellites orbiting above us. By the end, you will be able to calculate how fast a satellite needs to go to stay in orbit, why astronauts feel weightless, and what escape velocity really means.

Kepler's Laws of Planetary Motion

Before Newton, Kepler used Tycho Brahe's detailed observations to figure out how planets move. Three laws: First law (Law of Orbits): planets move in elliptical orbits with the Sun at one focus. Not circles - ellipses. Second law (Law of Areas): a line joining a planet to the Sun sweeps out equal areas in equal times. A planet moves faster when closer to the Sun (perihelion) and slower when farther (aphelion). Third law (Law of Periods): the square of the orbital period T is proportional to the cube of the semi-major axis a: T^2 proportional to a^3. For circular orbits: T^2 = (4 pi^2/GM_Sun) r^3. Kepler's laws are empirical - they describe what happens. Newton later explained WHY using his law of gravitation.

Newton's Universal Law of Gravitation

Every particle in the universe attracts every other particle with a force proportional to the product of their masses and inversely proportional to the square of the distance: F = G m_1 m_2 / r^2. G = 6.67 x 10^-11 N-m^2/kg^2 - the universal gravitational constant, same everywhere. The force is along the line joining the masses, always attractive. This law is universal - applies to apples and atoms and galaxies. The inverse square means doubling distance reduces force to ONE FOURTH. The Moon is 60 Earth radii away, so it feels 1/3600 of the gravitational pull that surface objects feel.

Acceleration Due to Gravity (g)

On Earth's surface, g = GM_E / R_E^2 = 9.8 m/s^2. But g varies: it decreases with height: g_h = g(1 - 2h/R) for h much less than R. It decreases with depth: g_d = g(1 - d/R). At Earth's center, g = 0 - equal mass pulling in all directions cancels. g also varies with latitude due to Earth's rotation - at the equator, centrifugal effects reduce g slightly (about 0.03 m/s^2 less than at poles). At poles g - 9.83 m/s^2, at equator g - 9.78 m/s^2. Earth's shape (oblate spheroid) also contributes. These variations matter for precise measurements and satellite launches.

Gravitational Potential and Potential Energy

Gravitational potential V at a point is the work per unit mass to bring a test mass from infinity: V = -GM/r. Zero at infinity, negative elsewhere. Gravitational potential energy U = -GMm/r (for two masses separated by r). Negative because gravity is attractive - you need to add energy to separate them. On Earth's surface, mgh is an approximation valid only for small heights. The general formula U = -GMm/r is always correct and becomes important for satellites, escape velocity, and celestial mechanics.

Escape Velocity

Escape velocity is the minimum speed to escape Earth's gravity forever - to reach infinity with zero KE remaining. v_esc = sqrt(2GM_E/R_E) = sqrt(2gR_E) - 11.2 km/s. Compare: orbital velocity in LEO is about 7.9 km/s. Escape velocity depends only on planet's mass and radius, not on the mass of the escaping object. The Moon's escape velocity is only 2.4 km/s - that is why it cannot hold an atmosphere. Gas molecules at room temperature move fast enough to escape lunar gravity. Earth's escape velocity keeps our atmosphere intact.

Satellites - Orbital Velocity and Time Period

A satellite in circular orbit has centripetal force provided by gravity: mv^2/r = GMm/r^2. Orbital velocity v = sqrt(GM/r). Depends only on central body's mass and orbital radius - not on satellite's mass. Closer satellites must move faster - LEO at 200 km requires about 7.8 km/s. Geostationary satellites orbit at 35,786 km above equator with period 24 hours - they appear fixed, perfect for communications. Time period T = 2 pi sqrt(r^3/GM). Kepler's third law in newtonian form. Energy of satellite: total E = -GMm/(2r) = KE + PE. Negative total energy means bound to Earth. To move to higher orbit, add energy.

Weightlessness

Astronauts in orbit appear weightless. But Earth's gravity at ISS altitude (~400 km) is still about 90% of surface gravity! So why do they float? Because they are in free fall. The spacecraft and everything inside fall toward Earth at the same acceleration. There is no normal reaction force pushing up - and that is what we perceive as weight. Weightless does not mean no gravity - it means no contact force. You experience the same momentarily on a roller coaster during a drop. True weightlessness requires being far from any massive body or being in continuous free fall.

Gravitational Field and Intensity

The gravitational field is the region around a mass where another mass experiences gravitational force. Field intensity E = F/m = GM/r^2. It is a vector pointing toward the mass. For a point mass, field intensity is just g at that point. The field is conservative - work done moving a mass around a closed path is zero. Field lines radiate inward. For a spherical mass, the field outside is as if all mass were at center (Shell theorem). Inside a uniform spherical shell, field is zero. Inside a solid sphere, field varies linearly: g = GMr/R^3 (for r < R).

Variation of g with Latitude and Rotation

Earth's rotation affects apparent weight. Centrifugal acceleration = omega^2 R cos^2 phi, where phi is latitude. At the equator, this reduces g by about 0.034 m/s^2. Effective g at latitude phi: g_eff = g - R omega^2 cos^2 phi. Things weigh slightly less at equator than at poles. Also Earth bulges at equator (equatorial radius ~21 km larger than polar), further reducing g at equator. For exam problems: g_eff = g_0 - R omega^2 cos^2 phi. Mumbai (near equator) has measurably lower g than Srinagar (north).

Key Points

  • •Kepler's laws: elliptical orbits, equal areas in equal times, T^2 proportional to a^3.
  • •Newton's law: F = G m_1 m_2 / r^2. G = 6.67 x 10^-11 N-m^2/kg^2. Universal attractive force.
  • •g = GM/R^2 = 9.8 m/s^2. Varies with height: g_h = g(1 - 2h/R). With depth: g_d = g(1 - d/R).
  • •At Earth's center, g = 0. At poles g - 9.83, at equator g - 9.78 m/s^2.
  • •Gravitational PE: U = -GMm/r. Zero at infinity, negative elsewhere.
  • •Escape velocity: v_esc = sqrt(2GM/R) = sqrt(2gR) = 11.2 km/s. Independent of escaping mass.
  • •Orbital velocity: v = sqrt(GM/r). Closer = faster. LEO ~7.9 km/s.
  • •Geostationary orbit: r - 42,000 km from Earth center, T = 24 h, above equator.
  • •Weightlessness = free fall. No normal reaction. Gravity is still present.
  • •Total energy of satellite: E = -GMm/(2r). Add energy ? higher orbit.
  • •Shell theorem: outside sphere, gravity as if all mass at center. Inside shell, gravity zero.
  • •Variation due to rotation: g_eff = g_0 - R omega^2 cos^2 phi. Equator has lowest g.
  • •Gravitational field is conservative - work around closed path is zero.

Practice Questions

  • State Kepler's laws. Derive Newton's law of gravitation from Kepler's third law.
  • Derive expression for g at height h above and depth d below Earth's surface.
  • Calculate escape velocity from Earth: R = 6400 km, g = 9.8 m/s^2.
  • A satellite orbits Earth at height 500 km. Find orbital velocity and time period. (R_E = 6400 km, M_E = 6 x 10^24 kg)
  • Why do astronauts in orbiting space station feel weightless despite gravity still acting?
  • Derive variation of g with latitude. Why is g slightly less at equator than poles?
  • What is a geostationary satellite? Calculate height of geostationary orbit.
  • State shell theorem for gravitational force. What is its significance?