Meccanica del punto materiale/Formulario

Cinematica

Moto rettilineo

$\begin{array}{cc}& a(t)=\text{ cost}\\ & v(t)=v_0+at\\ & s(t)=s_0+v_{0}t+\frac{1}{2}a{t}^2\end{array}$

Moto circolare

$\begin{array}{cc}& \omega =\frac{d\phi }{dt}=\frac{v}{r}\omega =2\pi \nu =\frac{2\pi }{T}\\ & a_c=\frac{v^2}{r}={\omega }^{2}r\end{array}$

Moto parabolico

$\begin{array}{cc}& x(t)=(v_0\cos {\theta }_0)t\\ & y(t)=(v_0\sin {\theta }_0)t-\frac{1}{2}g{t}^2\\ & x_{\text{gittata}}=\sin (2{\theta }_0)\frac{v_0^2}{g}\\ & y(x)=\tan {\theta }_{0}x-\frac{g}{2v_0^2{\cos }^2{\theta }_0}\\ & y_{\text{quota max}}=\frac{3}{2}{\sin }^2{\theta }_0\frac{v_0^2}{g}\\ \end{array}$

Trasformazioni di Galileo

$\{\begin{array}{cc}& t^{\prime }=t\\ & x^{\prime }=x-v_{0x}t\\ & y^{\prime }=y-v_{0y}t\\ & z^{\prime }=z-z_{0x}t\end{array}\{\begin{array}{cc}& v{\text{'}}_x=v_x-v_{0x}\\ & v{\text{'}}_y=v_y-v_{0y}\\ & v{\text{'}}_x=v_z-v_{0z}\end{array}\{\begin{array}{cc}& a{\text{'}}_x=a_x\\ & a{\text{'}}_y=a_y\\ & a{\text{'}}_z=a_z\end{array}$

Dinamica

${\overrightarrow{f}}_{tot}=\sum _i^n{\overrightarrow{f}}_i=ma$

Piano inclinato

$\begin{array}{cc}& a=g\sin \theta \\ & v=(g\sin \theta )t+v_0\\ & s=x_0+v_{0}t+\frac{1}{2}(g\sin \theta )t^2\end{array}L=\frac{h}{\sin \theta }\{\begin{array}{cc}& v_0=0\\ & s_0=0\end{array}\begin{array}{cc}& t=\sqrt{\frac{2h}{g{\sin }^2\theta }}\\ & v=g\sin \theta \sqrt{\frac{2h}{g{\sin }^2\theta }}\end{array}$

Quantità di moto e impulso

$\overrightarrow{q}=m\overrightarrow{v}$

$\overrightarrow{f}=m\overrightarrow{a}=m\frac{d\overrightarrow{v}}{dt}=\frac{dm\overrightarrow{v}}{dt}=\frac{d\overrightarrow{q}}{dt}$

$\overrightarrow{I}=\underset{t_0}{\overset{t}{\int }}fdt<\overrightarrow{f}>\mathrm{\Delta }t=\mathrm{\Delta }\overrightarrow{q}$

Momento della forza e momento angolare

$\overrightarrow{\tau }=\overrightarrow{r}\wedge \overrightarrow{f}$

$\overrightarrow{j}=\overrightarrow{r}\wedge \overrightarrow{q}=\overrightarrow{r}\wedge m\overrightarrow{v}$

$\overrightarrow{\tau }=\frac{d\overrightarrow{j}}{dt}$

Pendolo

$\frac{d^2\theta }{d{t}^2}=-\frac{g}{l}\sin \theta (t)$

$\theta (t)=A\sin (\omega t+\phi )\omega =\sqrt{\frac{g}{l}}T=\sqrt{\frac{l}{g}}2\pi$

Attrito

$F<{\mu }_{s}NF={\mu }_{d}N$

Forza elastica

$f=-kx$

Energia e Lavoro

$L=\underset{A}{\overset{B}{\int }}\overrightarrow{f}\cdot d\overrightarrow{s}$

$K=\frac{1}{2}m{v}^{2}L=\mathrm{\Delta }K$

Pendolo con lavoro

$v_{max}^2=2lg(1-\cos {\theta }_0)$

Moto di una molla

$\frac{d^{2}x}{d{t}^2}+\frac{k}{m}x=0$

$x(t)=A\sin (\omega t+\phi )\omega =\sqrt{\frac{k}{m}}$

$v_{max}=\mathrm{\Delta }\sqrt{\frac{k}{m}}=A\omega$

Energia potenziale

$U_{peso}=mgh{U}_{molla}=\frac{1}{2}k{x}^2{U}_{grav}=-G\frac{mM}{r}$

$\overrightarrow{f}=-\mathrm{\nabla }U\{\begin{array}{cc}& f_x=-\frac{\partial U}{\partial x}\\ & f_y=-\frac{\partial U}{\partial y}\\ & f_{<}=-\frac{\partial u}{\partial z}\end{array}$

Oscillatori armonici

Smorzato

$\frac{d^{2}x}{d{t}^2}+\frac{\mathrm{B}}{m}\frac{dx}{dt}+\frac{k}{m}x=0$

$\gamma =\frac{\mathrm{B}}{2m}{\omega }_0^2=\frac{k}{m}$

$x(t)=x_0{e}^{-\gamma t}\sin (\omega t)$

$\left|\frac{\mathrm{\Delta }E}{E}\right|=2\gamma T$

Forzato

$\frac{d^{2}x}{d{t}^2}+2\gamma \frac{dx}{dt}+{\omega }_0^{2}x=A\sin (\mathrm{\Omega }t)$

$x(t)=x_0\sin (\mathrm{\Omega }t-\delta )$

$\{\begin{array}{cc}& x_0=\frac{A}{\sqrt{({\omega }_0^2-{\mathrm{\Omega }}^2{)}^2+4\gamma {\mathrm{\Omega }}^2}}\\ & \delta =\arctan \left(\frac{2\gamma \mathrm{\Omega }}{{\omega }_0^2-{\mathrm{\Omega }}^2}\right)\end{array}$