Meccanica del punto materiale
Copertina Tutti i moduli · Sviluppo
Introduzione
Cinematica del punto materiale
Dinamica del punto materiale
Moti relativi
Lavoro ed energia
Oscillatori armonici
Appendice
Modifica il sommario
a ( t ) = cost v ( t ) = v 0 + a t s ( t ) = s 0 + v 0 t + 1 2 a t 2 {\displaystyle {\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}}}
ω = d φ d t = v r ω = 2 π ν = 2 π T a c = v 2 r = ω 2 r {\displaystyle {\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}}}
x ( t ) = ( v 0 cos θ 0 ) t y ( t ) = ( v 0 sin θ 0 ) t − 1 2 g t 2 x gittata = sin ( 2 θ 0 ) v 0 2 g y ( x ) = tan θ 0 x − g 2 v 0 2 cos 2 θ 0 y quota max = 3 2 sin 2 θ 0 v 0 2 g {\displaystyle {\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}}}
{ t ′ = t x ′ = x − v 0 x t y ′ = y − v 0 y t z ′ = z − z 0 x t { v x ′ = v x − v 0 x v y ′ = v y − v 0 y v x ′ = v z − v 0 z { a x ′ = a x a y ′ = a y a z ′ = a z {\displaystyle {\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}}}
f → t o t = ∑ i n f → i = m a {\displaystyle {\vec {f}}_{tot}=\sum _{i}^{n}{\vec {f}}_{i}=ma}
a = g sin θ v = ( g sin θ ) t + v 0 s = x 0 + v 0 t + 1 2 ( g sin θ ) t 2 L = h sin θ { v 0 = 0 s 0 = 0 t = 2 h g sin 2 θ v = g sin θ 2 h g sin 2 θ {\displaystyle {\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}}}
q → = m v → {\displaystyle {\vec {q}}=m{\vec {v}}}
f → = m a → = m d v → d t = d m v → d t = d q → d t {\displaystyle {\vec {f}}=m{\vec {a}}=m{\frac {d{\vec {v}}}{dt}}={\frac {dm{\vec {v}}}{dt}}={\frac {d{\vec {q}}}{dt}}}
I → = ∫ t 0 t f d t < f → > Δ t = Δ q → {\displaystyle {\vec {I}}=\int _{t_{0}}^{t}fdt\quad <{\vec {f}}>\Delta t=\Delta {\vec {q}}}
τ → = r → ∧ f → {\displaystyle {\vec {\tau }}={\vec {r}}\wedge {\vec {f}}}
j → = r → ∧ q → = r → ∧ m v → {\displaystyle {\vec {j}}={\vec {r}}\wedge {\vec {q}}={\vec {r}}\wedge m{\vec {v}}}
τ → = d j → d t {\displaystyle {\vec {\tau }}={\frac {d{\vec {j}}}{dt}}}
d 2 θ d t 2 = − g l sin θ ( t ) {\displaystyle {\frac {d^{2}\theta }{dt^{2}}}=-{\frac {g}{l}}\sin \theta (t)}
θ ( t ) = A sin ( ω t + φ ) ω = g l T = l g 2 π {\displaystyle \theta (t)=A\sin(\omega t+\varphi )\quad \omega ={\sqrt {\frac {g}{l}}}\quad T={\sqrt {\frac {l}{g}}}2\pi }
F < μ s N F = μ d N {\displaystyle F<\mu _{s}N\quad F=\mu _{d}N}
f = − k x {\displaystyle f=-kx}
L = ∫ A B f → ⋅ d s → {\displaystyle L=\int _{A}^{B}{\vec {f}}\cdot d{\vec {s}}}
K = 1 2 m v 2 L = Δ K {\displaystyle K={\frac {1}{2}}mv^{2}\quad L=\Delta K}
v m a x 2 = 2 l g ( 1 − cos θ 0 ) {\displaystyle v_{max}^{2}=2lg(1-\cos \theta _{0})}
d 2 x d t 2 + k m x = 0 {\displaystyle {\frac {d^{2}x}{dt^{2}}}+{\frac {k}{m}}x=0}
x ( t ) = A sin ( ω t + φ ) ω = k m {\displaystyle x(t)=A\sin(\omega t+\varphi )\quad \omega ={\sqrt {\frac {k}{m}}}}
v m a x = Δ k m = A ω {\displaystyle v_{max}=\Delta {\sqrt {\frac {k}{m}}}=A\omega }
U p e s o = m g h U m o l l a = 1 2 k x 2 U g r a v = − G m M r {\displaystyle U_{peso}=mgh\quad U_{molla}={\frac {1}{2}}kx^{2}\quad U_{grav}=-G{\frac {mM}{r}}}
f → = − ∇ U { f x = − ∂ U ∂ x f y = − ∂ U ∂ y f < = − ∂ u ∂ z {\displaystyle {\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}}}
d 2 x d t 2 + B m d x d t + k m x = 0 {\displaystyle {\frac {d^{2}x}{dt^{2}}}+{\frac {\mathrm {B} }{m}}{\frac {dx}{dt}}+{\frac {k}{m}}x=0}
γ = B 2 m ω 0 2 = k m {\displaystyle \gamma ={\frac {\mathrm {B} }{2m}}\quad \omega _{0}^{2}={\frac {k}{m}}}
x ( t ) = x 0 e − γ t sin ( ω t ) {\displaystyle x(t)=x_{0}\,e^{-\gamma t}\sin(\omega t)}
| Δ E E | = 2 γ T {\displaystyle \left|{\frac {\Delta E}{E}}\right|=2\gamma T}
d 2 x d t 2 + 2 γ d x d t + ω 0 2 x = A sin ( Ω t ) {\displaystyle {\frac {d^{2}x}{dt^{2}}}+2\gamma {\frac {dx}{dt}}+\omega _{0}^{2}x=A\sin(\Omega t)}
x ( t ) = x 0 sin ( Ω t − δ ) {\displaystyle x(t)=x_{0}\sin(\Omega t-\delta )}
{ x 0 = A ( ω 0 2 − Ω 2 ) 2 + 4 γ Ω 2 δ = arctan ( 2 γ Ω ω 0 2 − Ω 2 ) {\displaystyle {\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}}}
