labaphys.md
# Лаба 213
$$\frac{d^2x}{dt^2}+2\beta\frac{dx}{dt}+\omega^2x=0$$
$$x(t)=x_0e^{-\beta t}\cos(\omega t + \phi _0)$$
$$\omega = \sqrt{\omega _0^2 - \beta _0^2}$$
$$\frac{d^2I}{dt^2}+\frac{R}{L}\frac{dI}{dt}+\frac{I}{LC}=0$$
$$\beta = \frac{R}{2L}$$
$$\omega _0 = \sqrt{\frac{1}{LC}}$$
$$\omega = \sqrt{\frac{1}{LC} - \frac{R^2}{4L^2}}$$
$$\lambda = \ln \frac{I(t)}{I(t + T)}=\ln \frac{e^{-\beta t}}{e^{-\beta (t + T)}}=\ln e^{\beta T} = \beta T$$
$$Q=\frac{\pi}{\lambda}$$
# Лаба 12
$$T=2 \pi \sqrt{\frac{J}{mgl}}$$
$$T=2 \pi \sqrt{\frac{J_0 + J_g}{(m_0 + m_g)gl}}$$
$$J_0 = \frac{(m_g + m_0)glT^2}{4\pi^2} - J_g$$
$$l = \frac{m_gL}{m_g + m_0}$$
$$J_0 = \frac{m_ggLT^2}{4\pi^2} - J_g$$
$$J_g=J_c + m_gL^2=m_g(\frac{r^2}{2}+L^2)$$
$$J_0 = \frac{m_ggLT^2}{4\pi^2} - m_g(\frac{r^2}{2}+L^2)$$