JEE Physics Gravitation Notes Set 04

Newton's Law of Gravitation

Every two particles attract each other with a force along the line joining them, proportional to the product of their masses and inversely proportional to the square of their separation: \( F=\frac{Gm_1m_2}{r^2} \), where \( G=6.67\times10^{-11}\ \text{N m}^2\text{kg}^{-2} \).
A uniform solid sphere or thin shell attracts an outside particle as if its whole mass were at the centre. The force is conservative, central and always attractive, and gravitational forces on a body from several masses add as vectors.

Acceleration due to Gravity

At the surface \( g=\frac{GM}{R^2} \) (about 9.8 m/s²). Variation:
1. At height h: \( g_h=g\frac{R^2}{(R+h)^2}\approx g\left(1-\frac{2h}R\right) \) for small h.
2. At depth d (uniform earth): \( g_d=g\left(1-\frac dR\right) \); zero at the centre.
3. Due to rotation at latitude λ: \( g'=g-\omega^2R\cos^2\lambda \), so g is least at the equator and equals the true value at the poles.
4. In terms of density, \( g=\tfrac43\pi G\rho R \).

Gravitational Field and Potential

Field strength is force per unit mass, \( \vec E=\vec F/m \), and potential is work per unit mass to bring a body from infinity, \( V=-\frac{GM}r \) for a point mass. They are related by \( \vec E=-\nabla V \).
1. Thin shell: field outside \( GM/r^2 \), field inside zero, potential inside constant \( -GM/R \).
2. Solid sphere: outside as a point mass; inside \( E=\frac{GMr}{R^3} \) and \( V=-\frac{GM(3R^2-r^2)}{2R^3} \), so the potential at the centre is \( -\frac{3GM}{2R} \).
3. Ring on its axis: \( E=\frac{GMx}{(R^2+x^2)^{3/2}} \), \( V=-\frac{GM}{\sqrt{R^2+x^2}} \).
4. Uniform disc on its axis: \( V=-\frac{2GM}{R^2}\left[\sqrt{R^2+x^2}-x\right] \).
5. Field of a body with a spherical cavity is found by superposition: the full sphere plus a negative-mass sphere filling the cavity.

Gravitational Potential Energy

The potential energy of two masses is \( U=-\frac{Gm_1m_2}r \). For a system of particles it is the sum over all pairs, for example three equal masses m at the corners of a triangle of side a have \( U=-\frac{3Gm^2}a \). The self energy of a uniform sphere is \( -\frac{3GM^2}{5R} \) and of a thin shell \( -\frac{GM^2}{2R} \).
Near the earth's surface, \( \Delta U=mgh \) is the small-height approximation of \( mgh\,/\,(1+h/R) \).

Escape Speed and Satellites

Escape speed from radius r: \( v_e=\sqrt{\frac{2GM}r}=\sqrt{2gR} \) (11.2 km/s for the earth), independent of the mass and direction of projection. For a body projected with speed \( kv_e \) (k < 1) the maximum distance from the centre is \( R/(1-k^2) \).
For a circular orbit of radius r: \( v_o=\sqrt{\frac{GM}r} \), \( T=2\pi\sqrt{\frac{r^3}{GM}} \), \( KE=\frac{GMm}{2r} \), \( PE=-\frac{GMm}r \), \( E=-\frac{GMm}{2r} \). Thus \( v_e=\sqrt2\,v_o \); a satellite in a lower orbit has more speed and kinetic energy but less total energy.
A satellite close to the earth has a period of about 84 minutes. A geostationary satellite orbits at about 36000 km above the surface (radius about 42000 km) in the equatorial plane from west to east with a period of 24 hours. Objects inside an orbiting satellite are in free fall and feel apparent weightlessness.

Kepler's Laws

1. Law of orbits: every planet moves in an ellipse with the Sun at one focus.
2. Law of areas: the line joining planet and Sun sweeps equal areas in equal times, because angular momentum is conserved; areal velocity \( =\frac L{2m} \).
3. Law of periods: \( T^2\propto a^3 \) where a is the semi-major axis; in a circular orbit \( T^2=\frac{4\pi^2}{GM}r^3 \).
In an elliptical orbit the planet is fastest at perihelion; \( v_1r_1=v_2r_2 \) at the two extreme points; total energy \( E=-\frac{GMm}{2a} \) is constant while KE and PE vary.

Binary Stars

Two stars of masses \( M_1 \) and \( M_2 \) at separation d revolve about their centre of mass with the same angular speed and period. Their orbital radii satisfy \( M_1R_1=M_2R_2 \), so the lighter star moves in the larger circle. The period is \( T=2\pi\sqrt{\frac{d^3}{G(M_1+M_2)}} \). Energy needed to separate them to infinity equals the magnitude of the total (negative) energy.