﻿ All the formulas related to simple Harmonic Motion

# 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

$$\omega = \sqrt {\frac{mgd}{I}}$$ $$T = \frac{2 \pi}{\omega} = 2 \pi \frac{I}{mgd}$$