Gravitation

In this chapter: Kepler's three laws, Newton's universal law of gravitation, the constant G, acceleration due to gravity and its variation with height and depth, gravitational potential energy and potential, escape speed, Earth satellites and the energy of an orbiting satellite.

Kepler's laws

From the planetary observations of Tycho Brahe, Kepler found three laws:

• Law of orbits: all planets move in elliptical orbits with the Sun at one of the foci.
• Law of areas: the line joining a planet to the Sun sweeps out equal areas in equal intervals of time. So a planet moves fastest at perihelion (nearest point) and slowest at aphelion (farthest point).
• Law of periods: the square of the time period of revolution is proportional to the cube of the semi-major axis of the ellipse: T² ∝ a³.

The law of areas is a direct result of conservation of angular momentum. Gravity is a central force, directed along the line from planet to Sun, so it exerts no torque about the Sun. The areal velocity is dA/dt = L/2m, which stays constant.

Universal law of gravitation

Every body in the universe attracts every other body with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them:

• The force acts along the line joining the two bodies, is always attractive, and forms an action-reaction pair.
• For several masses, the net force is the vector sum of the individual forces (superposition).
• A uniform spherical shell attracts an outside point mass as if all its mass were at its centre, and exerts no net force on a point mass anywhere inside it. A solid sphere behaves the same way for outside points.
• G is a universal constant. It is very small, which is why the attraction between everyday objects goes unnoticed.

Acceleration due to gravity

For a body of mass m on the Earth's surface, mg = GMm/R², so

Variation with height and depth

At depth d, the shell of Earth above the point exerts no net force, so only the inner sphere of radius (R_E − d) pulls on the body. Assuming uniform density, g falls linearly to zero at the centre. So g is greatest at the surface and decreases both above and below it.

Gravitational potential energy

The formula mgh works only near the surface, where g is nearly constant. In general, taking the potential energy to be zero at infinity, the potential energy of mass m at a distance r (≥ R_E) from the Earth's centre is

For a system of particles, the total potential energy is the sum over all pairs, −Gm_im_j/r_ij. Gravity is a conservative force, so the work done in moving a mass depends only on the initial and final positions.

Escape speed

The minimum speed with which a body must be projected from the Earth's surface so that it never returns is the escape speed. Setting the total energy equal to zero, ½mv² − GM_Em/R_E = 0:

Worked example: Find the escape speed from the Earth, taking g = 9.8 m s⁻² and R_E = 6.4 × 10⁶ m.
Solution: v_e = √(2 × 9.8 × 6.4 × 10⁶) = √(1.254 × 10⁸) ≈ 1.12 × 10⁴ m s⁻¹ = 11.2 km s⁻¹. For the Moon, with a much smaller g and radius, it is only about a fifth of the Earth's value, which is why the Moon has no atmosphere.

Earth satellites

For a satellite in a circular orbit of radius r = R_E + h, gravity provides the centripetal force: mv²/r = GM_Em/r².

The orbital speed does not depend on the mass of the satellite; a heavier satellite at the same height moves at the same speed. A satellite in a higher orbit moves more slowly and takes longer to go round. A satellite in an equatorial orbit with a period of 24 hours, moving in the same sense as the Earth's rotation, appears fixed above one point on the Earth; such geostationary satellites orbit at a height of about 36 000 km and are used for communication.

Energy of an orbiting satellite

Worked example: A 400 kg satellite is in a circular orbit of radius 2R_E. How much energy is needed to shift it to a circular orbit of radius 4R_E? (g = 9.81 m s⁻², R_E = 6.37 × 10⁶ m)
Solution: ΔE = E_f − E_i = −GMm/(8R_E) + GMm/(4R_E) = GMm/(8R_E). Using GM = gR_E², ΔE = mgR_E/8 = 400 × 9.81 × 6.37 × 10⁶/8 ≈ 3.12 × 10⁹ J. In the new orbit its kinetic energy decreases by this amount and its potential energy increases by twice this amount.

Common mistakes: (1) Using mgh for heights comparable to the Earth's radius; use −GMm/r. (2) Measuring r from the Earth's surface instead of its centre in orbit and gravity formulas. (3) Thinking g is zero in orbit; astronauts feel weightless because they and their spacecraft are falling freely together, not because gravity vanishes. (4) Writing that the escape speed depends on the angle of projection or on the mass of the body. (5) Using g(1 − 2h/R) when h is not small compared with R.

JEE and NEET focus

• Variation of g with height, depth and (for JEE) the shape and rotation of the Earth.
• Ratios: escape speed or g for a planet with a different radius or density.
• Orbital speed, period and the energy relations KE : PE : E = 1 : −2 : −1.
• Kepler's third law in ratio form and the law of areas as angular momentum conservation.
• Energy needed to shift a satellite between orbits or to launch it from the surface.

Practice questions