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
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F = \frac{Gm_1m_2}{r^2}}

Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} . with two larger masses placed next to it. The gravitational force put torque on the axial spring, causing a rotation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum \tau = -k \Delta \varphi + 2F_Gr = 0} (1)

Mirror connected to axial spring would rotate and reflect light against a wall Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} meters away. The distance the light moved is represented by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 \Delta \varphi = \frac{h}{L}} (2)

Solve Equation 1 for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U=mgy} only works if gravitation force is constant. In terms of Gravitational forces:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U = -\frac{GmM}{r}}

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

Example: Escape Velocity

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2}mv_0^2-\frac{GmM}{R}=\lim_{r\to\infty} \frac{GmM}{r}} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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