PHYS 218 Chapter 12

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Gravitational Force

• An attractive force between all particles in the universe.
• These forces can be superimposed, added, etc.
• Only for point-like or spherical particles (can be approximated when size of particles is relatively small compared to the distance between them)

Force between two point-like or spherical particles
$F=\frac{G{m}_1{m}_2}{r^2}$

Where $G=6.67\times 10^{-11}$ is the gravitational constant (an extremely small number measured in Nm²/kg²)

Calculating value of G

Cavendish Experiment

Axial spring makes torque when turned (τ = k Δ ϕ). Two small masses connected to rod of length $r$. with two larger masses placed next to it. The gravitational force put torque on the axial spring, causing a rotation:

$\sum \tau =-k\mathrm{\Delta }\phi +2F_{G}r=0$ (1)

Mirror connected to axial spring would rotate and reflect light against a wall $L$ meters away. The distance the light moved is represented by $h$

$2\mathrm{\Delta }\phi =\frac{h}{L}$ (2)

Solve Equation 1 for $F_G$

Mystery 1: Dark Energy

Because of Gravitational Forces, any exploding body would expect to slow down and converge again.

On the contrary, galaxies are moving farther apart... and speeding up (74% of universe)

Mystery 2: Dark Matter

We should be able to calculate the tangential velocity of each planet based on what we've learned in this class. On a galactic scale, the stars actually move faster than they should... Is there a "filler mass" between the stars? (23% of universe)

Potential Energy

$U=mgy$ only works if gravitation force is constant. In terms of Gravitational forces:

$U=-\frac{GmM}{r}$

Notice that it's always negative, but do not confuse with the force.

Example: Escape Velocity

$\frac{1}{2}m{v}_0^2-\frac{GmM}{R}=\underset{r\to \mathrm{\infty }}{\lim }\frac{GmM}{r}$ $v_0=\sqrt{\frac{2GM}{R}}$

Kepler's Laws: Motion of Satellites

(can be derived from Netwon's Laws)

1. Each planet moves in an elliptic orbit with the sun at one focus point
2. A line from the sun to a planet sweeps equal area in equal times (change in area over time is constant and equivalent to angular momentum over twice the mass: $\frac{\mathrm{\Delta }A}{\mathrm{\Delta }t}=\frac{L}{2m}$
3. Period is proportional to (the major axis)^3/2

Note: In a binary star system, the masses are so huge that the center of mass is the focus of the orbital ellipse