JEE Physics Gravitation Notes Set 03

Newton's Law and Gravitational Field

Every two particles attract with a force \(F = \frac{Gm_1m_2}{r^2}\) along the line joining them, where \(G = 6.67\times 10^{-11}\ \text{N m}^2\text{kg}^{-2}\). The field strength (force per unit mass) at distance \(r\) from a point mass \(M\) is \(g = \frac{GM}{r^2}\). For fields and potentials of many masses, add the fields as vectors and the potentials as plain numbers.

Acceleration due to Gravity

Outside a uniform solid sphere or shell the field falls as \(\frac{1}{r^2}\). Inside a solid sphere it grows linearly with \(r\), and inside a thin shell it is zero. The field is maximum at the surface. If the earth shrinks with the same mass, \(g \propto \frac{1}{R^2}\) and a 1 percent decrease in \(R\) raises \(g\) by about 2 percent.

Potential and Potential Energy

1. Potential at distance \(r\) from a point mass: \(V = -\frac{GM}{r}\). Potential energy of a pair: \(U = -\frac{GMm}{r}\).
2. Field and potential are related by \(g = -\frac{dV}{dr}\). The field is perpendicular to equipotential surfaces and equals \(\frac{\Delta V}{\Delta d}\) across a perpendicular spacing \(\Delta d\).
3. Gravity is conservative, so the work done between two points is path independent and equals the change in potential energy.
4. Raising a body from the surface to height \(h\): \(\Delta U = GMm\left(\frac{1}{R} - \frac{1}{R+h}\right) = \frac{mgh}{1 + h/R}\). The formula \(mgh\) holds only for \(h \ll R\).
5. For a body at the centre of a uniform earth, \(g = 0\) and so the weight is zero.

Escape Velocity and Orbital Motion

1. Escape velocity: \(v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR}\). It does not depend on the mass of the projectile or on the direction of projection. Orbital speed near the surface is \(v_o = \sqrt{gR} = \frac{v_e}{\sqrt{2}}\).
2. A satellite at orbital radius \(r\): \(v = \sqrt{\frac{GM}{r}}\), \(T = 2\pi\sqrt{\frac{r^3}{GM}}\). Neither depends on the satellite's mass.
3. Energies of a circular orbit: \(KE = \frac{GMm}{2r}\), \(PE = -\frac{GMm}{r}\), \(E = -\frac{GMm}{2r}\). The total energy is negative for every bound orbit, and the energy to escape from it equals \(\frac{GMm}{2r}\).
4. Geostationary satellite: period 24 h, height about 6R above the surface (radius about 7R), orbit in the equatorial plane. All such satellites have the same radius.
5. Projected with speed \(kv_e\) (\(k<1\)) vertically: the maximum distance from the centre is \(\frac{R}{1-k^2}\).

Kepler's Laws

1. Orbits are ellipses with the sun at one focus.
2. The line joining the planet to the sun sweeps equal areas in equal times, because angular momentum \(mvr\) is conserved. The planet is fastest at the nearest point and slowest at the farthest point.
3. \(T^2 \propto a^3\), where \(a\) is the semi-major axis. If the force varies as \(r^{-n}\), then \(T^2 \propto r^{n+1}\).
4. For small changes, \(\frac{\Delta T}{T} = \frac{3}{2}\frac{\Delta R}{R}\).

Worked Examples

1. Find the height at which \(g\) falls to one fourth of its surface value: \(\left(\frac{R}{R+h}\right)^2 = \frac{1}{4}\) gives \(R + h = 2R\), so \(h = R\).
2. A satellite orbits at \(r = 4R\). Its speed is \(\sqrt{\frac{GM}{4R}} = \frac{1}{2}\sqrt{gR}\) and its period is \(2\pi\sqrt{\frac{64R^3}{GM}} = 8\times T_0\), where \(T_0\) is the period of a near-surface satellite.
3. Two fixed masses \(m\) a distance \(l\) apart: the potential at the mid-point is \(-\frac{4Gm}{l}\) per unit mass, so a particle needs \(v = \sqrt{\frac{8Gm}{l}}\) to escape from there.
4. A planet is in an elliptical orbit with nearest and farthest distances \(d\) and \(3d\). Then \(v_{near} = 3v_{far}\) and \(KE_{near} = 9\,KE_{far}\).

Common Mistakes

1. Using the height instead of the distance from the centre in the inverse square law.
2. Applying \(mgh\) for heights comparable with the earth's radius.
3. Forgetting that bound orbits have negative total energy.
4. Treating \(g\) as zero inside a solid sphere. It is zero only at the centre, and zero everywhere inside a thin shell.