Gravitation

Newton's law of universal gravitation

$$F_{grav} = \frac{Gm_1m_2}{r^2}$$

• where $r>r_1$ and $r>r_2$ (i.e. two bodies outside of each other)
• $G = 6.67\times10^{-11}$ N$\cdot$m$^2$/kg$^2$ : gravitational constant

Gravitational PE (away from planet surface)

$$U_{grav} = -\frac{GMm}{r}$$

• If $r = \infty$, $U_{grav} = 0$
• As $U_{grav} < 0$, $U_{grav}$ decreases (more negative) as $r$ decreases

Shell theorem

Outside a hollow sphere ($r \ge R$)

$$\begin{align*} F_{shell} &= \frac{GMm}{r^2} \\ U_{shell} &= -\frac{GMm}{r} \end{align*}$$

Inside a hollow sphere ($r \lt R$)

$$\begin{align*} F_{shell} &= 0 \\ U_{shell} &= -\frac{GMm}{R} \end{align*}$$

• $R$: radius of hollow sphere
• $M$: mass of hollow sphere

Gravitational force inside a sphere of uniform density

$$F = \frac{GMm}{R^3}r$$

Weight

Acceleration due to gravity:

$$g_0 = \frac{GM}{R^2}$$

True weight at the poles:

$$w_0 = mg_0$$

Apparent weight at the equator:

$$w = mg_0 - \frac{mv^2}{R}$$

• $M$: mass of the planet
• $R$: radius of the planet

(Minimum) escape speed

If a projectile is launched vertically, ignoring air resistance

$$\begin{align*} \text{When }v&\rightarrow0\text{ and }r\rightarrow\infty \\ \frac{1}{2}mv^2 &+ \left(-\frac{GMm}{R}\right) = 0 + 0\\ \therefore v_{esc} &= \sqrt{\frac{2GM}{R}} \end{align*}$$

Schwarzschild radius $r_s$ :

$$\begin{align*} c = \sqrt{\frac{2GM}{r_s}} &&\rightarrow&& r_s = \frac{2GM}{c^2} \end{align*}$$

Satellite circular orbit

$$\begin{align*} \frac{mv^2}{r} &= \frac{GMm}{r^2} \\ \therefore v_{circ} &= \sqrt{\frac{GM}{r}} \\ T &= \frac{2\pi r}{v} = \frac{2\pi r^{\frac{3}{2}}}{\sqrt{GM}} \\ E_{orbit} &= \frac{U}{2} = -\frac{GMm}{2r} \end{align*}$$

For a larger orbit,

• Total energy and PE are larger (less negative)
• But KE is smaller

Kepler's laws of planetary motion

First law: elliptical orbit around the Sun

$$\begin{align*} \text{Equation:} && \frac{x^2}{a^2} &+ \frac{y^2}{b^2} = 1 \\ \text{Foci:} && \pm c &= \sqrt{a^2 - b^2} \\ \text{Eccentricity:} && e &= \sqrt{1 - \frac{b^2}{a^2}} \\ \text{Aphelion:} && r_{max} &= a(1+e) \\ \text{Perihelion:} && r_{min} &= a(1-e) \\ \end{align*}$$

• $a$: length of semi-major axis
• $b$: length of semi-minor axis
• Aphelion is the farthest point from Sun
• Perihelion is the closest point to Sun

Second law: constant sector velocity

$$\begin{align*} \frac{dA}{dt} &= \frac{1}{2}r^2\frac{d\theta}{dt} \\ &= \frac{1}{2M}Mvr\sin\theta \\ &= \frac{L}{2M} \end{align*}$$

• $A$: area swept by the planet in time $t$
• $L$: angular momentum of the planet which is constant, ignoring precession

Third law: period of elliptical orbit

$$\begin{align*} T = \frac{2\pi a^{\frac{3}{2}}}{\sqrt{GM_s}} \end{align*}$$

• $a$: length of semi-major axis
• $M_s$: mass of Sun