List of formulas related to simple Harmonic Motion
Spring
K: spring constant (N/m), s: displacement (m), ω: angular velocity (rad/s), T: time period, m: mass of the object, A: amplitude
$$ s = Acos(\omega t + \phi) $$
$$ \omega = \sqrt{ \frac{k}{m}} $$
$$ T = \frac{1}{f} = \frac{2 \pi}{\omega} = 2 \pi \sqrt{ \frac{m}{k}} $$
Springs in Series
k: spring constant of the system, T: time period of the system
$$ \frac{1}{k} = \frac{1}{k_1} + \frac{1}{k_2} + ... + \frac{1}{k_n} $$
$$ T^2 = T_1 ^2 + T_2 ^2$$
Springs in Parallel
$$ k = k_1 + k_2 $$
$$ \frac{1}{T^2} = \frac{1}{T_1 ^2} + \frac{1}{T_2 ^2} $$
Simple Pendulum
ω: angular velocity (rad/s), T: time period , αo: angle at amplitude, A: amplitude, l: length of the pendulum
$$ \omega = \frac{g}{l} $$
$$ T = \frac{2 \pi}{\omega} = 2 \pi \frac{l}{g} $$
$$ \alpha _o = \frac{A}{l} $$
Motion Equations
S: displacement, α: angle, v: velocity, T: strength on the rod
$$ s = Acos(\omega t + \phi) $$
$$ \alpha = \alpha _o cos(\omega t + \phi) $$
$$ v = \sqrt{2gl(cos \alpha - cos \alpha _o)} $$
$$ T = mg(3 cos \alpha - 2 cos \alpha _o) $$
Energy
Ep: potential energy, Ek: kinetic energy, E: mechanical energy
$$ E_p = mgl(1 - cos \alpha) $$
$$ E_k = \frac{1}{2}m v^2 $$
$$ E = E_p + E_k = \frac{1}{2} mgl \alpha _o ^2 $$
Change of Period Following The Change of The Temperature
Α: coefficient of linear expansion of the rod
$$ \Delta t^o = t_2 ^o - t_1 ^o $$
$$ \frac{\Delta T}{T} = \frac{\alpha}{2} \Delta t^o $$
Change of Period Following The Change of The Height
h: height of the pendulum, R: earth’s radius
$$ \frac{\Delta T}{T} = \frac{h}{R} $$
Pendulum
I: moment of inertia, ω: angular velocity, m: mass of bob, d: OG, T: time period
$$ \omega = \sqrt {\frac{mgd}{I}} $$
$$ T = \frac{2 \pi}{\omega} = 2 \pi \frac{I}{mgd} $$
