---
# System prepended metadata

title: Resonance
tags: [mechanics, SHM]

---

# Resonance
when $\omega = \omega_0$ resonance will occur
$$F = kx_{m0}$$

$$F_0cos(\omega t) = m\ddot x+kx$$

$$x_{m0} = \frac Fk$$

$$\omega_0^2 = \frac km$$
:::success
$$x_{m0} = \frac{F_0}{m\omega_0^2}$$
:::
## Power
$$P(t) = FV = F_0cos(\omega t)(-x_{m0}\omega_0sin(\omega t))$$

$$= F_0cos(\omega t)(-x_{m0}\omega_0sin(\omega t))$$

$$P(t) = \frac12x_{m0}\omega_0F_0sin(\omega t)$$
:::success
$$P = \frac{1}{T}\int_0^T P(t) dt = 0$$
:::
## Energy
$$\frac 12kx_{m0}^2$$