Meccanica del punto materiale/Formulario

Cinematica

Moto rettilineo

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

Moto circolare

${\begin{aligned}&\omega ={\frac {d\varphi }{dt}}={\frac {v}{r}}\quad \omega =2\pi \nu ={\frac {2\pi }{T}}\\&a_{c}={\frac {v^{2}}{r}}=\omega ^{2}r\end{aligned}}$

Moto parabolico

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

Trasformazioni di Galileo

${\begin{cases}&t'=t\\&x'=x-v_{0x}t\\&y'=y-v_{0y}t\\&z'=z-z_{0x}t\end{cases}}\quad {\begin{cases}&v'_{x}=v_{x}-v_{0x}\\&v'_{y}=v_{y}-v_{0y}\\&v'_{x}=v_{z}-v_{0z}\\\end{cases}}\quad {\begin{cases}&a'_{x}=a_{x}\\&a'_{y}=a_{y}\\&a'_{z}=a_{z}\end{cases}}$

Dinamica

${\vec {f}}_{tot}=\sum _{i}^{n}{\vec {f}}_{i}=ma$

Piano inclinato

${\begin{aligned}&a=g\sin \theta \\&v=(g\sin \theta )t+v_{0}\\&s=x_{0}+v_{0}t+{\frac {1}{2}}(g\sin \theta )t^{2}\end{aligned}}\quad L={\frac {h}{\sin \theta }}\quad {\begin{cases}&v_{0}=0\\&s_{0}=0\end{cases}}\quad {\begin{aligned}&t={\sqrt {\frac {2h}{g\sin ^{2}\theta }}}\\&v=g\sin \theta {\sqrt {\frac {2h}{g\sin ^{2}\theta }}}\end{aligned}}$

Quantità di moto e impulso

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

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

${\vec {I}}=\int _{t_{0}}^{t}fdt\quad <{\vec {f}}>\Delta t=\Delta {\vec {q}}$

Momento della forza e momento angolare

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

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

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

Pendolo

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

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

Attrito

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

Forza elastica

$f=-kx$

Energia e Lavoro

$L=\int _{A}^{B}{\vec {f}}\cdot d{\vec {s}}$

$K={\frac {1}{2}}mv^{2}\quad L=\Delta K$

Pendolo con lavoro

$v_{max}^{2}=2lg(1-\cos \theta _{0})$

Moto di una molla

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

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

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

Energia potenziale

$U_{peso}=mgh\quad U_{molla}={\frac {1}{2}}kx^{2}\quad U_{grav}=-G{\frac {mM}{r}}$

${\vec {f}}=-\nabla U\quad {\begin{cases}&f_{x}=-{\frac {\partial U}{\partial x}}\\&f_{y}=-{\frac {\partial U}{\partial y}}\\&f_{<}=-{\frac {\partial u}{\partial z}}\end{cases}}$

Oscillatori armonici

Smorzato

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

$\gamma ={\frac {\mathrm {B} }{2m}}\quad \omega _{0}^{2}={\frac {k}{m}}$

$x(t)=x_{0}\,e^{-\gamma t}\sin(\omega t)$

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

Forzato

${\frac {d^{2}x}{dt^{2}}}+2\gamma {\frac {dx}{dt}}+\omega _{0}^{2}x=A\sin(\Omega t)$

$x(t)=x_{0}\sin(\Omega t-\delta )$

${\begin{cases}&x_{0}={\frac {A}{\sqrt {(\omega _{0}^{2}-\Omega ^{2})^{2}+4\gamma \Omega ^{2}}}}\\&\delta =\arctan \left({\frac {2\gamma \Omega }{\omega _{0}^{2}-\Omega ^{2}}}\right)\end{cases}}$