xSkill: notes, lectures and practice for CBSE and CSE

1.1 The universal law and Kepler Every particle attracts every other particle with a force 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 particles and is always attractive. The constant of proportionality is the universal gravitational constant G, which is the same everywhere. Kepler's three laws describe planetary motion. The orbit of a planet is an ellipse with the sun at one focus. The line from the sun to the planet sweeps equal areas in equal times, which is the statement of angular momentum conservation. The square of the period is proportional to the cube of the semi-major axis. Newton showed that the inverse square law produces exactly this third law.

F = \frac{G m _{1} m _{2}}{r ^{2}}, G = 6.67 \times 1 0^{- 11} N m^{2} kg^{- 2}

Newton's law of universal gravitation

T^{2} = \frac{4 π ^{2}}{GM} a^{3}

Kepler's third law for a planet of semi-major axis a around a mass M

1.2 Variation of g The acceleration due to gravity at the surface of the Earth follows from the law and the mass and radius of the Earth. Above the surface it falls as the inverse square of the distance from the centre, so g at a height h is smaller than the surface value. Below the surface, inside a uniform sphere, the enclosed mass falls as the cube of the distance, so g decreases linearly with depth and is zero at the centre. The value of g also changes with latitude because the Earth rotates, and it is slightly larger at the poles than at the equator. For school problems the Earth is treated as a uniform sphere and the rotation effect is ignored unless the question mentions it.

g_{h} = \frac{GM}{( R + h ) ^{2}}, g_{d} = g (1 - \frac{d}{R})

g at a height h above the surface and at a depth d below it

1.3 Field, potential and energy The gravitational field at a point is the force per unit mass on a small test mass, directed toward the attracting body. The gravitational potential is the work per unit mass needed to bring a small mass from infinity to that point, with zero at infinity, so it is negative everywhere near a mass. The potential energy of a pair of masses is the negative of G m1 m2 over their separation, which is why a bound system has negative total energy. The field is the negative gradient of the potential, so the field points from higher to lower potential. For a spherical body, the field and the potential outside are the same as if the whole mass were at the centre. Inside a uniform shell the field is zero, which is a result of the shell theorem.

V = - \frac{GM}{r}, U = - \frac{G m _{1} m _{2}}{r}

Gravitational potential and potential energy, with zero at infinity

1.4 Escape and orbital speeds The escape speed is the minimum speed with which a body can leave the Earth and never return, found by setting the total energy to zero: the square root of twice G M over R, which is about 11.2 km per second for the Earth. The orbital speed of a satellite close to the surface is the square root of G M over R, which is about 7.9 km per second, smaller than the escape speed by a factor of root two. A satellite in a circular orbit of radius r has orbital speed root of G M over r and period 2 pi r over that speed, which recovers Kepler's third law. Its kinetic energy is half the magnitude of its potential energy, and its total energy is negative; the extra energy needed to escape from the orbit is the negative of this total energy.

v_{e} = \frac{2 GM}{R}, v_{o} = \frac{GM}{r}, E = - \frac{GM m}{2 r}

Escape speed, orbital speed and the total energy of a circular orbit

Exam

Keep the signs of potential energy negative. A bound orbit has negative total energy, and the escape condition is total energy equal to zero, not positive kinetic energy.