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 {Gm_{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:

$\sum \tau =-k\Delta \varphi +2F_{G}r=0$ (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 $h$

$2\Delta \varphi ={\frac {h}{L}}$ (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

${\frac {1}{2}}mv_{0}^{2}-{\frac {GmM}{R}}=\lim _{r\to \infty }{\frac {GmM}{r}}$ $v_{0}={\sqrt {\frac {2GM}{R}}}$

Kepler's Laws: Motion of Satellites

(can be derived from Netwon's Laws)

Each planet moves in an elliptic orbit with the sun at one focus point
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: ${\frac {\Delta A}{\Delta t}}={\frac {L}{2m}}$
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