Kinematics

$  v = u + at $
$  v^2 = u^2 + 2as $
$  s=ut + \frac{1}{2}at^2 $

Measurement

If $ A = B \pm C $
$ \Delta A = \Delta B + \Delta C $
If $ A = BC^k $
$ \frac{\Delta A}{A} = \frac{\Delta B}{B} + k\frac{\Delta C}{C} $

Forces and Dynamics

$ F = kx $

For springs in series:
$ k_{T} = \frac{1}{k_{1} + k_{2} + … + k_{n}} $

For springs in parallel:
$ k_{t} = k_{1}+k_{2}+ … + k_{n} $

$ p = \rho gh $
$ U = m_{fluid.displaced}g $
$\quad = V_{fluid.displaced}\rho g $

$ F = \frac{d(mv)}{dt} $
$\quad = m \frac{dv}{dt} + v \frac{dm}{dt}$

Special cases:
if $ \frac{dv}{dt} = 0,$
$\quad F = v \frac{dm}{dt}$

if $ \frac{dm}{dt} = 0,$
$\quad F = m\frac{dv}{dt} = ma$

$ F = \rho Av^2 $

Derivation:
$ F = \frac{d(mv)}{dt} $
$ \quad = \frac{d(\rho Vv)}{dt} $

if v is constant,
$ \quad = v\frac{d(\rho V)}{dt} $
$ \quad = v\frac{d(\rho Ax)}{dt} $
$ \quad = \rho Av\frac{d(x)}{dt} $
$ \quad = \rho Av^2 $

Work, Energy and Power

$ PE = mgh $
$ KE = \frac{1}{2}mv^2 $
$ EPE = \frac{1}{2}kx^2 $
$ P = Fv $

Circular Motion

$ a = \frac{v^2}{r} = r\omega^2$
$ F = \frac{mv^2}{r} = mr\omega^2$

Gravitational Field

$ F = G\frac{Mm}{r^2} $
$ g = G\frac{M}{r^2} $
$ U = -G\frac{Mm}{r} $
$ \phi = -G\frac{M}{r} $
$ F = -\frac{dU}{dr} $
$ g = -\frac{d\phi}{dr} $

Thermal Physics

$ PV = nRT $
$ T/K = T/C^\circ + 273.15 $
$ PV = \frac{1}{3} Nm \langle c \rangle ^2 $
$ \frac{1}{2} m \langle c \rangle ^2 = \frac{3}{2} kT $

First Law of Thermodynamics

$ \Delta U = Q_{in} + W_{on} $
$W = P \Delta V $

Oscillations

$ a = -\omega^2x $ for both

Derivation:
$ x = x_{o}sin(\omega t) $ or $ x = x_{o}cos(\omega t) $

Differentiating with respect to time,
$ v = v_{o}\omega cos(\omega t) $ or $ v= -v_{o}\omega sin(\omega t) $

Differentiating with respect to time again,
$ a = -v_{o}\omega ^ 2 sin(\omega t) $ or $ a = -v_{o}\omega ^2 cos(\omega t) $
or
$ a = -\omega^2x $ for both

$ v = \pm \omega \sqrt{x_{o}^2-x^2} $

$ KE = \frac{1}{2}m\omega ({x_{o}^2-x^2}) $
$ TE = \frac{1}{2}m\omega {x_{o}^2} $
$ PE = TE – KE = \frac{1}{2}m\omega {x^2} $

Wave Motion

$ v = f \lambda $
$ I \propto A^2 $
$ I \propto cos^2 \theta $

Superposition

$ \Delta x = \frac{\lambda D}{a} $
$ sin \theta = \frac{\lambda}{b} $
$ \theta \approx \frac{\lambda}{b} $
$ dsin\theta = n\lambda $

Electric Fields

$ F = \frac{1}{4\pi \varepsilon_{0}} \frac{Q_{1}Q_{2}}{R^2} $
$ E = \frac{1}{4\pi \varepsilon_{0}} \frac{Q}{R^2} $
$ U = \frac{1}{4\pi \varepsilon_{0}} \frac{Q_{1}Q_{2}}{R} $
$ V = \frac{1}{4\pi \varepsilon_{0}} \frac{Q}{R} $
$ E = \frac{V}{D} $

Current of Electricity

$ Q = It $
$ V = \frac{W}{Q} $
$ V = IR $
$ P = VI $ or $ P = I^2R $ or $ P = \frac{V^2}{R} $
$ R = \frac{\rho l}{A} $

D.C. Circuits

For resistors in series:
$ R_{T} = R_{1} + R_{2} + … + R_{n} $

For resistors in parallel:
$ R_{t} = \frac{1}{R_{1}+R_{2}+ … + R_{n}} $
Potential divider rule:
$ V_{1}= \frac{R_{1}} {R_{1} + R_{2} + … + R_{n}} \times V_{t} $

Electromagnetism

$ B = \frac{\mu_{o} I}{2\pi d} $
$ B = \frac{\mu_{o} nI}{2r} $
$ B = \mu_{o} nI $
$ F = BIl sin \theta $
$ F = Bqv sin \theta $

Electromagnetic Induction

$ \Phi = BA cos \theta $
$ E = \frac{d\Phi}{dt} $

Alternating Current

$ I_{rms} = \frac{I_{o}}{\sqrt{2}} $
$ \frac{N_s}{N_p} = \frac{V_s}{V_p} = \frac{I_p}{I_s} $

Quantum Physics

$ \frac{1}{2}mv_{max}^2 = eV_{s} $
$ hf = \Phi + \frac{1}{2}mv_{max}^2 $
$ \lambda = \frac{h}{\rho} $
$ hf = E_{2} – E_{1} $
$ \Delta p \Delta x \geq h $

Nuclear Physics

$ E = mc^2 $
$ t_{\frac{1}{2}} = \frac{ln2}{\lambda} $