User:Buckleyj

For EC103, Assignment 2 1A source on: Node 1: $$\frac{v_2 - v_1}{10} + 1A + \frac{2v_x - v_1}{15} = \frac{v_1}{5}$$ Leads to $$15v_1 - 7v_2 = 30$$

Node 2: $$\frac{2v_2 - 2v_1 - v_2}{10} = \frac{v_2 - v_1}{10} + \frac{2v_2}{10}$$ Leads to $$v_1 + 2v_2 = 0$$

2A source on: Node 1: $$\frac{v_2 - v_1}{10} + \frac{2v_2 - 2v_1 - v_1}{15} = \frac{v_1}{5}$$ Leads to $$7v_2 - 15v_1 = 0$$

Node 2: $$\frac{2v_2 - 2v_1 - v_2}{10} + 2A = \frac{2v_2}{10} + \frac{v_2 - v_1}{10}$$ Leads to $$2v_2 + v_1 = 20A$$

For 620-143 Assignment 2 $$B(m) = \int_{0}^{1} x^{m-1} (1-x)^{\frac{-1}{2}} dx$$

For physics cheat sheet

Movement

Average velocity: $$v_{average} = \frac{s}{t}$$

Acceleration: $$a = \frac{ \Delta\ v}{t} = \frac{v-u}{t}$$

Force due to gravity: $$F_g = mg$$

Momentum: $$p = mv$$

Impulse: $$F \Delta\ t = m \Delta\ v$$

Work: $$W = Fs = \Delta\ Energy$$

Kinetic energy: $$K_e = \frac{1}{2}mv^2$$

Gravitational potential energy: $$GP_e = mgh$$

Force from a spring: $$F = -kx$$

Elastic potential energy: $$EP_e = \frac{1}{2}kx^2$$

Velocity around a circle: $$v =\frac{2 \pi\ r}{T}$$

Acceleration around a circle: $$a = \frac{v^2}{r} = \frac{4 \pi\ ^2r}{T^2}$$

Centripetal force: $$F_c = \frac{mv^2}{r}$$

Universal gravitation: (F is in $$NKg^{-1}$$) $$F = G\frac{M_1M_2}{r^2}$$

Gravitational field strength: $$F = G\frac{M}{r^2}$$

Orbits around the same mass: $$\frac{T_1^2}{T_2^2} = \frac{r_1^3}{r_2^3}$$

Electronics

Work: $$W = Vq = VIt$$

Power: $$P = VI = I^2R = \frac{V^2}{R}$$

Voltage: $$V = IR$$

Votage divider: $$V_{out} = V_{in} \frac{R_2}{R_1 + R_2}$$

Voltage gain: $$G = \frac{ \Delta\ V_{out}}{ \Delta\ V_{in}}$$

Further Electronics

RMS: $$V_{RMS} = \frac{V_{peak}}{\sqrt{2}}$$

$$I_{RMS} = \frac{I_{peak}}{\sqrt{2}}$$

Transformer: $$\frac{N_p}{N_s} = \frac{V_p}{V_s} = \frac{I_s}{I_p}$$

$$P_{in} = P_{out}$$

Time constant: $$t = RC$$

Ripple voltage: $$V_{R(p-p)} = V_{max} - V_{min} = \frac{V_{max}T}{RC}$$

Light

Path difference for nodal lines: $$ p.d. = (n - \frac{1}{2}) \lambda\ $$

Path difference for antinodal lines: $$ p.d. = n \lambda\ $$

For the double slit experiment: $$w = \frac{ \lambda\ L}{d} = \frac{vL}{ \mathit{f} d}$$

For single slit experiment: $$sin \Theta\ = \frac{ \lambda\ }{w}$$

Energy of a photon: $$E = h \mathit{f} = \frac{hc}{ \lambda\ }$$

Number of photos emitted by a light source: $$E_{light source} = nE_{photon}$$

The photoelectric effect: $$h \mathit{f} = W + KE_{electron}$$

Photon Momentum: $$p = \frac{E}{c} = \frac{h \mathit{f}}{c} = \frac{ \mathit{f}}{ \lambda}$$

de Broglie Wavelength: $$\mathbf{de\ Broglie\ wavelength}\ \lambda\ = \frac{h}{p} = \frac{h}{mv}$$

Spectra of atoms: $$2 \pi\ r = n \lambda\ $$

$$ \varepsilon\ = \frac{-N \Delta\ \phi\ }{ \Delta\ t} $$