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\frac{m}{s^2}"],["g = 9.81 \\frac{m}{s^2}"]),B=h(["g_{luna} = 1.62 \frac{m}{s^2}"],["g_{luna} = 1.62 \\frac{m}{s^2}"]),J=h(["g_{marte} = 3.71 \frac{m}{s^2}"],["g_{marte} = 3.71 \\frac{m}{s^2}"]),K=h(["left | \vec{F} \right | leq mu_{s} left | \vec{F}_{normale} \right |"],["\\left | \\vec{F} \\right | \\leq \\mu_{s} \\left | \\vec{F}_{normale} \\right |"]),Q=h(["left | \vec{F} \right | leq mu_{d} left | \vec{F}_{normale} \right |"],["\\left | \\vec{F} \\right | \\leq \\mu_{d} \\left | \\vec{F}_{normale} \\right |"]),X=h(["F = -k x"],["F = -k x"]),Y=h(["Delta \vec{s} = \vec{s}(fine) - \vec{s}(inizio)"],["\\Delta \\vec{s} = \\vec{s}(fine) - \\vec{s}(inizio)"]),$=h(["\vec{v} = \frac{Delta \vec{s}}{Delta t}"],["\\vec{v} = \\frac{\\Delta \\vec{s}}{\\Delta t}"]),Z=h(["\vec{v} = lim_{Delta t \to 0} \frac{Delta \vec{s}}{Delta t} = \frac{d \vec{s}}{dt}"],["\\vec{v} = \\lim_{\\Delta t \\to 0} \\frac{\\Delta \\vec{s}}{\\Delta t} = \\frac{d \\vec{s}}{dt}"]),ee=h(["\vec{a} = \frac{Delta 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x^2"],["E_{p_e} = \\frac{1}{2} k x^2"]),ze=h(["E = E_k + E_p"],["E = E_k + E_p"]),xe=h(["P = \frac{Delta E}{Delta t}"],["P = \\frac{\\Delta E}{\\Delta t}"]),Ee=String.raw,De=Object(g.h)("h1",null,"Fisica"),Se=Object(g.h)("h2",null,"Vettori"),Ce=Object(g.h)("h3",null,"Componenti cartesiane"),Fe=Object(g.h)("p",null,"Usa le regole base della trigonometria:"),Pe=Object(g.h)("h3",null,"Somma"),Te=Object(g.h)("p",null,"Scomponi in componenti, poi sommali:"),ke=Object(g.h)("p",null,"Produce il vettore risultante dall'applicazione della regola del parallelogramma."),Ne=Object(g.h)("h3",null,"Differenza"),Ae=Object(g.h)("p",null,"Alla fine è sempre una somma:"),Le=Object(g.h)("p",null,"Produce il vettore che parte da ",Object(g.h)(z,null,"w")," e arriva a ",Object(g.h)(z,null,"v"),"."),Me=Object(g.h)("h3",null,"Prodotto scalare"),Ue=Object(g.h)("p",null,"Si chiama scalare perchè il risultato è uno scalare, non un vettore."),We=Object(g.h)("h2",null,"Leggi di Newton"),Ve=Object(g.h)("h3",null,"1ᵃ: Inerzia"),qe=Object(g.h)("p",null,"Se un corpo puntiforme ha forza risultante nulla, allora la sua velocità non cambia."),Re=Object(g.h)("h3",null,"2ᵃ: Proporzionalità"),Ie=Object(g.h)("p",null,"La forza risultante di un corpo è direttamente proporzionale alla sua accelerazione, e la costante di proporzionalità è la ",Object(g.h)("i",null,"massa"),"."),He=Object(g.h)("h3",null,"3ᵃ: Azione e reazione"),Ge=Object(g.h)("p",null,"Due corpi esercitano forze uguali e opposte uno sull'altro."),Be=Object(g.h)("h2",null,"Forza di gravità"),Je=Object(g.h)("h3",null,"Tra due corpi"),Ke=Object(g.h)("p",null,"Due corpi puntiformi si attirano uno verso l'altro con forza:"),Qe=Object(g.h)("p",null,Object(g.h)(z,null,"G")," è la ",Object(g.h)("i",null,"costante di gravitazione universale")," e vale:"),Xe=Object(g.h)("h3",null,"Verso la Terra"),Ye=Object(g.h)("p",null,"Se nel sistema di riferimento consideriamo la Terra ferma, allora un corpo è attratto verso la Terra con forza ",Object(g.h)("i",null,"peso")," uguale a:"),$e=Object(g.h)("p",null,Object(g.h)(z,null,"g")," è la ",Object(g.h)("i",null,"costante di gravità")," della Terra, e vale:"),Ze=Object(g.h)("h3",null,"Su pianeti diversi"),et=Object(g.h)("p",null,"Per pianeti diversi dalla Terra vale la stessa regola:"),tt=Object(g.h)("p",null,"L'unica differenza è che cambia la ",Object(g.h)("i",null,"costante di gravità"),":"),nt=Object(g.h)("h2",null,"Forze di contatto"),lt=Object(g.h)(D,null,Object(g.h)("h3",null,"Normale"),Object(g.h)("p",null,"Si oppone alle forze applicate alla superficie di contatto."),Object(g.h)("p",null,"Un libro appoggiato su un tavolo ha la ",Object(g.h)("b",null,"forza di gravità")," che lo attira verso il terreno e la ",Object(g.h)("b",null,"forza normale")," che lo trattiene dal cadere.")),ot=Object(g.h)("h3",null,"Attrito statico"),ct=Object(g.h)("p",null,"Impedisce a un corpo di muoversi se non viene spinto da una forza che supera una certa soglia:"),rt=Object(g.h)("h3",null,"Attrito dinamico"),at=Object(g.h)("p",null,"Rallenta i corpi che si stanno muovendo finchè essi non si fermano:"),it=Object(g.h)(D,null,Object(g.h)("h3",null,"Tensione"),Object(g.h)("p",null,"E' forza trasmessa tra due estremi di una fune."),Object(g.h)("p",null,"Può essere redirezionata per mezzo di carrucole.")),ut=Object(g.h)("h3",null,"Elastica"),pt=Object(g.h)("p",null,"Una molla cerca sempre di tornare alla sua posizione indeformata con forza:"),ht=Object(g.h)("p",null,"(E' negativa perchè la forza è opposta a quella applicata per deformarla.)"),st=Object(g.h)("h2",null,"Cinematica"),bt=Object(g.h)("h3",null,"Spostamento"),ft=Object(g.h)("p",null,"È un vettore che indica la posizione di un corpo rispetto a un'origine."),Ot=Object(g.h)("h3",null,"Velocità"),vt=Object(g.h)("p",null,"È un vettore che misura la variazione di posizione nel tempo."),jt=Object(g.h)("p",null,"Se si considera un intervallo di tempo infinitesimale si dice ",Object(g.h)("i",null,"velocità istantanea"),":"),dt=Object(g.h)("h3",null,"Accelerazione"),mt=Object(g.h)("p",null,"È un vettore che misura la variazione di velocità nel tempo."),_t=Object(g.h)("p",null,"Se si considera un intervallo di tempo infinitesimale si dice ",Object(g.h)("i",null,"accelerazione istantanea"),":"),gt=Object(g.h)("h3",null,"Quantità di moto ",Object(g.h)("small",null,"(momento lineare)")),yt=Object(g.h)("p",null,"La quantità di moto è una proprietà vettoriale dei corpi:"),wt=Object(g.h)("p",null,"Se la forza risultante è nulla, la quantità di moto non cambia."),zt=Object(g.h)("h2",null,"Moto rettilineo uniforme"),xt=Object(g.h)("h3",null,"Spostamento"),Et=Object(g.h)("p",null,"La ",Object(g.h)("i",null,"legge oraria")," è:"),Dt=Object(g.h)("h3",null,"Velocità"),St=Object(g.h)("p",null,"È costante:"),Ct=Object(g.h)("h3",null,"Accelerazione"),Ft=Object(g.h)("p",null,"La velocità non varia:"),Pt=Object(g.h)(D,null,Object(g.h)("h3",null,"Forze"),Object(g.h)("p",null,"Si applica la prima legge di Newton:"),Object(g.h)("p",null,Object(g.h)(z,null,"f(t) = 0"))),Tt=Object(g.h)("h2",null,"Moto rettilineo uniformemente accelerato"),kt=Object(g.h)("h3",null,"Spostamento"),Nt=Object(g.h)("p",null,"La ",Object(g.h)("i",null,"legge oraria")," è:"),At=Object(g.h)("h3",null,"Velocità"),Lt=Object(g.h)("p",null,"È una retta:"),Mt=Object(g.h)("h3",null,"Accelerazione"),Ut=Object(g.h)("p",null,"È costante:"),Wt=Object(g.h)(D,null,Object(g.h)("h3",null,"Forze"),Object(g.h)("p",null,"Si applica la prima legge di Newton:"),Object(g.h)("p",null,Object(g.h)(z,null,"f(t) = m a"))),Vt=Object(g.h)("h2",null,"Moto armonico semplice"),qt=Object(g.h)(D,null,Object(g.h)("h3",null,"Ampiezza"),Object(g.h)("p",null,"E' la distanza dal centro massima che raggiunge il corpo."),Object(g.h)("p",null,"(L'ampiezza di una sinusoide.)")),Rt=Object(g.h)("h3",null,"Velocità angolare"),It=Object(g.h)("p",null,"Indica quanto in fretta cambia la posizione del corpo."),Ht=Object(g.h)("p",null,"Dipende dal periodo:"),Gt=Object(g.h)("h3",null,"Spostamento"),Bt=Object(g.h)("p",null,"E' una sinusoide:"),Jt=Object(g.h)("h3",null,"Velocità"),Kt=Object(g.h)("h3",null,"Accelerazione"),Qt=Object(g.h)(D,null,Object(g.h)("h3",null,"Forze"),Object(g.h)("p",null,"Si applica la prima legge di Newton:"),Object(g.h)("p",null,Object(g.h)(z,null,"f(t) = m a"))),Xt=Object(g.h)("h2",null,"Moti composti"),Yt=Object(g.h)(D,null,Object(g.h)("h3",null,"Moto parabolico"),Object(g.h)("p",null,"Il moto parabolico è dato sommando un moto rettilineo uniforme sull'asse orizzontale e un moto rettilineo uniformemente accelerato sull'asse verticale.")),$t=Object(g.h)("h3",null,"Moto circolare uniforme"),Zt=Object(g.h)("h2",null,"Moto circolare uniforme"),en=Object(g.h)("h3",null,"Velocità angolare"),tn=Object(g.h)("p",null,"Quanto cambia la fase nel tempo."),nn=Object(g.h)("h3",null,"Fase"),ln=Object(g.h)("p",null,"E' l'angolo percorso dal corpo rispetto alla posizione iniziale."),on=Object(g.h)("h3",null,"Velocità"),cn=Object(g.h)("p",null,"Si applicano le formule per la circonferenza:"),rn=Object(g.h)("h3",null,"Accelerazione"),an=Object(g.h)("p",null,"Il corpo ha sempre un accelerazione verso il centro che gli impedisce di abbandonare il moto:"),un=Object(g.h)("h3",null,"Forza centripeta"),pn=Object(g.h)("h3",null,"È verso il centro e si calcola con:"),hn=Object(g.h)("h2",null,"Lavoro ed energia"),sn=Object(g.h)("h3",null,"Lavoro"),bn=Object(g.h)("p",null,"E' compiuto da una forza che sposta un corpo."),fn=Object(g.h)("p",null,"(Se la forza non è parallela allo spostamento, il prodotto scalare ci fa considerare solo la componente parallela.)"),On=Object(g.h)("h3",null,"Energia cinetica"),vn=Object(g.h)("p",null,"Un corpo ha energia cinetica in ogni momento uguale a:"),jn=Object(g.h)("p",null,"Se una forza effettua lavoro su un 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\n

Fisica

\n

Vettori

\n \n \n

\n Componenti cartesiane\n

\n

\n Usa le regole base della trigonometria:\n

\n

\n {r`\\vec{v} = \\vec{v}_x + \\vec{v}_y`}\n

\n

\n {r`\\left | \\vec{v}_x \\right | = \\left | \\vec{v} \\right | \\sin \\alpha`}\n

\n

\n {r`\\left | \\vec{v}_y \\right | = \\left | \\vec{v} \\right | \\cos \\alpha`}\n

\n \n \n

\n Somma\n

\n

\n Scomponi in componenti, poi sommali:\n

\n

\n {r`\\vec{v} + \\vec{w} = (\\vec{v}_x + \\vec{w}_x) + (\\vec{v}_y + \\vec{w}_y)`}\n

\n

\n Produce il vettore risultante dall'applicazione della regola del parallelogramma.\n

\n \n \n

\n Differenza\n

\n

\n Alla fine è sempre una somma:\n

\n

\n {r`\\vec{v} - \\vec{w} = (\\vec{v}_x - \\vec{w}_x) + (\\vec{v}_y - \\vec{w}_y)`}\n

\n

\n Produce il vettore che parte da w e arriva a v.\n

\n \n \n

\n Prodotto scalare\n

\n

\n Si chiama scalare perchè il risultato è uno scalare, non un vettore.\n

\n

\n {r`\\vec{v} \\cdot \\vec{w} = \\left | \\vec{v} \\right | \\left | \\vec{w} \\right | \\cos \\alpha`}\n

\n

\n Produce il modulo della proiezione di {r`\\vec{a}`} su {r`\\vec{b}`}.\n

\n \n \n

\n Leggi di Newton\n

\n \n \n

\n 1ᵃ: Inerzia\n

\n

\n Se un corpo puntiforme ha forza risultante nulla, allora la sua velocità non cambia.\n

\n

\n {r`\\Sigma \\vec{F} = 0 \\Longleftrightarrow \\Delta v = 0`}\n

\n \n \n

\n 2ᵃ: Proporzionalità\n

\n

\n La forza risultante di un corpo è direttamente proporzionale alla sua accelerazione, e la costante di proporzionalità è la massa.\n

\n

\n {r`\\Sigma \\vec{F} = m \\vec{a}`}\n

\n \n \n

\n 3ᵃ: Azione e reazione\n

\n

\n Due corpi esercitano forze uguali e opposte uno sull'altro. \n

\n

\n {r`\\vec{F}_{21} = -\\vec{F}_{12}`}\n

\n \n \n

\n Forza di gravità\n

\n \n \n

\n Tra due corpi\n

\n

\n Due corpi puntiformi si attirano uno verso l'altro con forza:\n

\n

\n {r`\\left | \\vec{F} \\right | = G \\frac{m_1 m_2}{s^2}`}\n

\n

\n G è la costante di gravitazione universale e vale:\n

\n

\n {r`G = 6.67 \\cdot 10^{-11} \\frac{N m^2}{{kg}^2}`}\n

\n \n \n

\n Verso la Terra\n

\n

\n Se nel sistema di riferimento consideriamo la Terra ferma, allora un corpo è attratto verso la Terra con forza peso uguale a:\n

\n

\n {r`\\left | \\vec{F} \\right | = g m`}\n

\n

\n g è la costante di gravità della Terra, e vale:\n

\n

\n {r`g = 9.81 \\frac{m}{s^2}`}\n

\n \n \n

\n Su pianeti diversi\n

\n

\n Per pianeti diversi dalla Terra vale la stessa regola:\n

\n

\n {r`\\left | \\vec{F} \\right | = g m`}\n

\n

\n L'unica differenza è che cambia la costante di gravità:\n

\n

\n {r`g_{luna} = 1.62 \\frac{m}{s^2}`}\n

\n

\n {r`g_{marte} = 3.71 \\frac{m}{s^2}`}\n

\n \n \n

\n Forze di contatto\n

\n \n \n

\n Normale\n

\n

\n Si oppone alle forze applicate alla superficie di contatto.\n

\n

\n Un libro appoggiato su un tavolo ha la forza di gravità che lo attira verso il terreno e la forza normale che lo trattiene dal cadere. \n

\n \n \n

\n Attrito statico\n

\n

\n Impedisce a un corpo di muoversi se non viene spinto da una forza che supera una certa soglia:\n

\n

\n {r`\\left | \\vec{F} \\right | \\leq \\mu_{s} \\left | \\vec{F}_{normale} \\right |`}\n

\n \n \n

\n Attrito dinamico\n

\n

\n Rallenta i corpi che si stanno muovendo finchè essi non si fermano:\n

\n

\n {r`\\left | \\vec{F} \\right | \\leq \\mu_{d} \\left | \\vec{F}_{normale} \\right |`}\n

\n \n \n

\n Tensione\n

\n

\n E' forza trasmessa tra due estremi di una fune.\n

\n

\n Può essere redirezionata per mezzo di carrucole.\n

\n \n \n

\n Elastica\n

\n

\n Una molla cerca sempre di tornare alla sua posizione indeformata con forza:\n

\n

\n {r`F = -k x`}\n

\n

\n (E' negativa perchè la forza è opposta a quella applicata per deformarla.)\n

\n \n \n

\n Cinematica\n

\n \n \n

\n Spostamento\n

\n

\n È un vettore che indica la posizione di un corpo rispetto a un'origine.\n

\n

\n {r`\\Delta \\vec{s} = \\vec{s}(fine) - \\vec{s}(inizio)`}\n

\n \n \n

\n Velocità\n

\n

\n È un vettore che misura la variazione di posizione nel tempo.\n

\n

\n {r`\\vec{v} = \\frac{\\Delta \\vec{s}}{\\Delta t}`}\n

\n

\n Se si considera un intervallo di tempo infinitesimale si dice velocità istantanea:\n

\n

\n {r`\\vec{v} = \\lim_{\\Delta t \\to 0} \\frac{\\Delta \\vec{s}}{\\Delta t} = \\frac{d \\vec{s}}{dt}`}\n

\n \n \n

\n Accelerazione\n

\n

\n È un vettore che misura la variazione di velocità nel tempo.\n

\n

\n {r`\\vec{a} = \\frac{\\Delta \\vec{v}}{\\Delta t}`}\n

\n

\n Se si considera un intervallo di tempo infinitesimale si dice accelerazione istantanea:\n

\n

\n {r`\\vec{a} = \\lim_{\\Delta v \\to 0} \\frac{\\Delta \\vec{v}}{\\Delta t} = \\frac{d \\vec{v}}{d t} = \\frac{d^2 \\vec{s}}{d t^2}`}\n

\n \n \n

\n Quantità di moto (momento lineare)\n

\n

\n La quantità di moto è una proprietà vettoriale dei corpi:\n

\n

\n {r`\\vec{p} = m \\vec{v}`}\n

\n

\n Se la forza risultante è nulla, la quantità di moto non cambia.\n

\n

\n {r`\\Sigma \\vec{F} = 0 \\Longleftrightarrow \\Delta \\vec{p} = 0`}\n

\n \n \n

\n Moto rettilineo uniforme\n

\n \n \n

\n Spostamento\n

\n

\n La legge oraria è:\n

\n

\n {r`s(t) = v \\cdot \\Delta t + s(0)`}\n

\n \n \n

\n Velocità\n

\n

\n È costante:\n

\n

\n {r`v(t) = k`}\n

\n \n \n

\n Accelerazione\n

\n

\n La velocità non varia:\n

\n

\n {r`a(t) = 0`}\n

\n \n \n

\n Forze\n

\n

\n Si applica la prima legge di Newton:\n

\n

\n f(t) = 0\n

\n \n \n

\n Moto rettilineo uniformemente accelerato\n

\n \n \n

\n Spostamento\n

\n

\n La legge oraria è:\n

\n

\n {r`s(t) = \\frac{1}{2} a \\cdot (\\Delta t)^2 + v(0) \\cdot (\\Delta t) + s(0)`}\n

\n \n \n

\n Velocità\n

\n

\n È una retta:\n

\n

\n {r`v(t) = a \\Delta t + v(0)`}\n

\n \n \n

\n Accelerazione\n

\n

\n È costante:\n

\n

\n {r`a(t) = k`}\n

\n \n \n

\n Forze\n

\n

\n Si applica la prima legge di Newton:\n

\n

\n f(t) = m a\n

\n \n \n

\n Moto armonico semplice\n

\n \n \n

\n Ampiezza\n

\n

\n E' la distanza dal centro massima che raggiunge il corpo.\n

\n

\n (L'ampiezza di una sinusoide.)\n

\n \n \n

\n Velocità angolare\n

\n

\n Indica quanto in fretta cambia la posizione del corpo. \n

\n

\n Dipende dal periodo:\n

\n

\n {r`\\omega = \\frac{2 \\pi}{T}`}\n

\n \n \n

\n Spostamento\n

\n

\n E' una sinusoide:\n

\n

\n {r`s(t) = A \\sin (\\omega \\cdot t + \\phi)`}\n

\n \n \n

\n Velocità\n

\n

\n E' la sinusoide dello spostamento, sfasata di {r`\\frac{\\pi}{2}`}:\n

\n

\n {r`s(t) = A \\sin (\\omega \\cdot t + \\phi + \\frac{\\pi}{2})`}\n

\n \n \n

\n Accelerazione\n

\n

\n E' la sinusoide della velocità, sfasata di {r`\\frac{\\pi}{2}`}:\n

\n

\n {r`s(t) = A \\sin (\\omega \\cdot t + \\phi + \\pi)`}\n

\n \n \n

\n Forze\n

\n

\n Si applica la prima legge di Newton:\n

\n

\n f(t) = m a\n

\n \n \n

\n Moti composti\n

\n \n \n

\n Moto parabolico\n

\n

\n Il moto parabolico è dato sommando un moto rettilineo uniforme sull'asse orizzontale e un moto rettilineo uniformemente accelerato sull'asse verticale.\n

\n \n \n

\n Moto circolare uniforme\n

\n

\n Il moto parabolico è dato sommando due moti armonici semplici: uno sull'asse X, e l'altro, sfasato di {r`\\frac{\\pi}{2}`}, sull'asse Y.\n

\n \n \n

\n Moto circolare uniforme\n

\n \n \n

\n Velocità angolare\n

\n

\n Quanto cambia la fase nel tempo.\n

\n

\n {r`\\omega = \\frac{2 \\pi}{T}`}\n

\n \n \n

\n Fase\n

\n

\n E' l'angolo percorso dal corpo rispetto alla posizione iniziale.\n

\n

\n Si indica con {r`\\phi`}, e generalmente si usa in radianti.\n

\n \n \n

\n Velocità\n

\n

\n Si applicano le formule per la circonferenza:\n

\n

\n {r`v = \\frac{\\Delta s}{t} = \\frac{2 \\pi \\cdot r}{T}`}\n

\n \n \n

\n Accelerazione\n

\n

\n Il corpo ha sempre un accelerazione verso il centro che gli impedisce di abbandonare il moto: \n

\n

\n {r`a = \\frac{v^2}{r} = r \\cdot \\omega^2 = v \\cdot \\omega`}\n

\n \n \n

\n Forza centripeta\n

\n

\n È verso il centro e si calcola con:\n

\n

\n {r`F = m \\cdot a`}\n

\n \n \n

\n Lavoro ed energia\n

\n \n \n

\n Lavoro\n

\n

\n E' compiuto da una forza che sposta un corpo.\n

\n

\n {r`W = \\vec{F} \\cdot \\vec{s} = F \\cdot \\Delta s \\cdot cos(\\alpha )`}\n

\n

\n (Se la forza non è parallela allo spostamento, il prodotto scalare ci fa considerare solo la componente parallela.)\n

\n \n \n

\n Energia cinetica\n

\n

\n Un corpo ha energia cinetica in ogni momento uguale a:\n

\n

\n {r`E_c = \\frac{1}{2} m v^2`}\n

\n

\n Se una forza effettua lavoro su un corpo, cambia la sua energia cinetica pari al lavoro effettuato:\n

\n

\n {r`\\Delta E_c = W`}\n

\n \n \n

\n Energia potenziale gravitazionale\n

\n

\n Un corpo ha energia potenziale in ogni momento pari a: \n

\n

\n {r`E_{p_g} = m \\cdot g \\cdot h`}\n

\n

\n (Con h uguale a un altezza scelta come punto di riferimento.)\n

\n \n \n

\n Energia potenziale elastica\n

\n

\n Una molla ha sempre energia potenziale elastica pari a:\n

\n

\n {r`E_{p_e} = \\frac{1}{2} k x^2`}\n

\n \n \n

\n Forze conservative\n

\n

\n Sono conservative le forze per le quali il lavoro compiuto non dipende dal percorso seguito per andare dalla partenza all'arrivo.\n

\n

\n Ad esempio, è conservativa la forza di gravità, ma non è conservativa la forza di attrito.\n

\n

\n Se in un sistema ci sono solo forze conservative, allora l'energia meccanica totale si conserva:\n

\n

\n {r`E = E_k + E_p`}\n

\n \n \n

\n Potenza\n

\n

\n È la velocità di trasferimento di energia:\n

\n

\n {r`P = \\frac{\\Delta E}{\\Delta t}`}\n

\n \n \n
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Fisica

Vettori

Componenti cartesiane

Usa le regole base della trigonometria:

\vec{v} = \vec{v}_x + \vec{v}_y

\left | \vec{v}_x \right | = \left | \vec{v} \right | \sin \alpha

\left | \vec{v}_y \right | = \left | \vec{v} \right | \cos \alpha

Somma

Scomponi in componenti, poi sommali:

\vec{v} + \vec{w} = (\vec{v}_x + \vec{w}_x) + (\vec{v}_y + \vec{w}_y)

Produce il vettore risultante dall'applicazione della regola del parallelogramma.

Differenza

Alla fine è sempre una somma:

\vec{v} - \vec{w} = (\vec{v}_x - \vec{w}_x) + (\vec{v}_y - \vec{w}_y)

Produce il vettore che parte da w e arriva a v.

Prodotto scalare

Si chiama scalare perchè il risultato è uno scalare, non un vettore.

\vec{v} \cdot \vec{w} = \left | \vec{v} \right | \left | \vec{w} \right | \cos \alpha

Produce il modulo della proiezione di \vec{a} su \vec{b}.

Leggi di Newton

1ᵃ: Inerzia

Se un corpo puntiforme ha forza risultante nulla, allora la sua velocità non cambia.

\Sigma \vec{F} = 0 \Longleftrightarrow \Delta v = 0

2ᵃ: Proporzionalità

La forza risultante di un corpo è direttamente proporzionale alla sua accelerazione, e la costante di proporzionalità è la massa.

\Sigma \vec{F} = m \vec{a}

3ᵃ: Azione e reazione

Due corpi esercitano forze uguali e opposte uno sull'altro.

\vec{F}_{21} = -\vec{F}_{12}

Forza di gravità

Tra due corpi

Due corpi puntiformi si attirano uno verso l'altro con forza:

\left | \vec{F} \right | = G \frac{m_1 m_2}{s^2}

G è la costante di gravitazione universale e vale:

G = 6.67 \cdot 10^{-11} \frac{N m^2}{{kg}^2}

Verso la Terra

Se nel sistema di riferimento consideriamo la Terra ferma, allora un corpo è attratto verso la Terra con forza peso uguale a:

\left | \vec{F} \right | = g m

g è la costante di gravità della Terra, e vale:

g = 9.81 \frac{m}{s^2}

Su pianeti diversi

Per pianeti diversi dalla Terra vale la stessa regola:

\left | \vec{F} \right | = g m

L'unica differenza è che cambia la costante di gravità:

g_{luna} = 1.62 \frac{m}{s^2}

g_{marte} = 3.71 \frac{m}{s^2}

Forze di contatto

Normale

Si oppone alle forze applicate alla superficie di contatto.

Un libro appoggiato su un tavolo ha la forza di gravità che lo attira verso il terreno e la forza normale che lo trattiene dal cadere.

Attrito statico

Impedisce a un corpo di muoversi se non viene spinto da una forza che supera una certa soglia:

\left | \vec{F} \right | \leq \mu_{s} \left | \vec{F}_{normale} \right |

Attrito dinamico

Rallenta i corpi che si stanno muovendo finchè essi non si fermano:

\left | \vec{F} \right | \leq \mu_{d} \left | \vec{F}_{normale} \right |

Tensione

E' forza trasmessa tra due estremi di una fune.

Può essere redirezionata per mezzo di carrucole.

Elastica

Una molla cerca sempre di tornare alla sua posizione indeformata con forza:

F = -k x

(E' negativa perchè la forza è opposta a quella applicata per deformarla.)

Cinematica

Spostamento

È un vettore che indica la posizione di un corpo rispetto a un'origine.

\Delta \vec{s} = \vec{s}(fine) - \vec{s}(inizio)

Velocità

È un vettore che misura la variazione di posizione nel tempo.

\vec{v} = \frac{\Delta \vec{s}}{\Delta t}

Se si considera un intervallo di tempo infinitesimale si dice velocità istantanea:

\vec{v} = \lim_{\Delta t \to 0} \frac{\Delta \vec{s}}{\Delta t} = \frac{d \vec{s}}{dt}

Accelerazione

È un vettore che misura la variazione di velocità nel tempo.

\vec{a} = \frac{\Delta \vec{v}}{\Delta t}

Se si considera un intervallo di tempo infinitesimale si dice accelerazione istantanea:

\vec{a} = \lim_{\Delta v \to 0} \frac{\Delta \vec{v}}{\Delta t} = \frac{d \vec{v}}{d t} = \frac{d^2 \vec{s}}{d t^2}

Quantità di moto (momento lineare)

La quantità di moto è una proprietà vettoriale dei corpi:

\vec{p} = m \vec{v}

Se la forza risultante è nulla, la quantità di moto non cambia.

\Sigma \vec{F} = 0 \Longleftrightarrow \Delta \vec{p} = 0

Moto rettilineo uniforme

Spostamento

La legge oraria è:

s(t) = v \cdot \Delta t + s(0)

Velocità

È costante:

v(t) = k

Accelerazione

La velocità non varia:

a(t) = 0

Forze

Si applica la prima legge di Newton:

f(t) = 0

Moto rettilineo uniformemente accelerato

Spostamento

La legge oraria è:

s(t) = \frac{1}{2} a \cdot (\Delta t)^2 + v(0) \cdot (\Delta t) + s(0)

Velocità

È una retta:

v(t) = a \Delta t + v(0)

Accelerazione

È costante:

a(t) = k

Forze

Si applica la prima legge di Newton:

f(t) = m a

Moto armonico semplice

Ampiezza

E' la distanza dal centro massima che raggiunge il corpo.

(L'ampiezza di una sinusoide.)

Velocità angolare

Indica quanto in fretta cambia la posizione del corpo.

Dipende dal periodo:

\omega = \frac{2 \pi}{T}

Spostamento

E' una sinusoide:

s(t) = A \sin (\omega \cdot t + \phi)

Velocità

E' la sinusoide dello spostamento, sfasata di \frac{\pi}{2}:

s(t) = A \sin (\omega \cdot t + \phi + \frac{\pi}{2})

Accelerazione

E' la sinusoide della velocità, sfasata di \frac{\pi}{2}:

s(t) = A \sin (\omega \cdot t + \phi + \pi)

Forze

Si applica la prima legge di Newton:

f(t) = m a

Moti composti

Moto parabolico

Il moto parabolico è dato sommando un moto rettilineo uniforme sull'asse orizzontale e un moto rettilineo uniformemente accelerato sull'asse verticale.

Moto circolare uniforme

Il moto parabolico è dato sommando due moti armonici semplici: uno sull'asse X, e l'altro, sfasato di \frac{\pi}{2}, sull'asse Y.

Moto circolare uniforme

Velocità angolare

Quanto cambia la fase nel tempo.

\omega = \frac{2 \pi}{T}

Fase

E' l'angolo percorso dal corpo rispetto alla posizione iniziale.

Si indica con \phi, e generalmente si usa in radianti.

Velocità

Si applicano le formule per la circonferenza:

v = \frac{\Delta s}{t} = \frac{2 \pi \cdot r}{T}

Accelerazione

Il corpo ha sempre un accelerazione verso il centro che gli impedisce di abbandonare il moto:

a = \frac{v^2}{r} = r \cdot \omega^2 = v \cdot \omega

Forza centripeta

È verso il centro e si calcola con:

F = m \cdot a

Lavoro ed energia

Lavoro

E' compiuto da una forza che sposta un corpo.

W = \vec{F} \cdot \vec{s} = F \cdot \Delta s \cdot cos(\alpha )

(Se la forza non è parallela allo spostamento, il prodotto scalare ci fa considerare solo la componente parallela.)

Energia cinetica

Un corpo ha energia cinetica in ogni momento uguale a:

E_c = \frac{1}{2} m v^2

Se una forza effettua lavoro su un corpo, cambia la sua energia cinetica pari al lavoro effettuato:

\Delta E_c = W

Energia potenziale gravitazionale

Un corpo ha energia potenziale in ogni momento pari a:

E_{p_g} = m \cdot g \cdot h

(Con h uguale a un altezza scelta come punto di riferimento.)

Energia potenziale elastica

Una molla ha sempre energia potenziale elastica pari a:

E_{p_e} = \frac{1}{2} k x^2

Forze conservative

Sono conservative le forze per le quali il lavoro compiuto non dipende dal percorso seguito per andare dalla partenza all'arrivo.

Ad esempio, è conservativa la forza di gravità, ma non è conservativa la forza di attrito.

Se in un sistema ci sono solo forze conservative, allora l'energia meccanica totale si conserva:

E = E_k + E_p

Potenza

È la velocità di trasferimento di energia:

P = \frac{\Delta E}{\Delta t}

© 2019 - Stefano Pigozzi - CC BY-SA 4.0
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\n

Fisica

\n

Vettori

\n \n \n

\n Componenti cartesiane\n

\n

\n Usa le regole base della trigonometria:\n

\n

\n {r`\\vec{v} = \\vec{v}_x + \\vec{v}_y`}\n

\n

\n {r`\\left | \\vec{v}_x \\right | = \\left | \\vec{v} \\right | \\sin \\alpha`}\n

\n

\n {r`\\left | \\vec{v}_y \\right | = \\left | \\vec{v} \\right | \\cos \\alpha`}\n

\n \n \n

\n Somma\n

\n

\n Scomponi in componenti, poi sommali:\n

\n

\n {r`\\vec{v} + \\vec{w} = (\\vec{v}_x + \\vec{w}_x) + (\\vec{v}_y + \\vec{w}_y)`}\n

\n

\n Produce il vettore risultante dall'applicazione della regola del parallelogramma.\n

\n \n \n

\n Differenza\n

\n

\n Alla fine è sempre una somma:\n

\n

\n {r`\\vec{v} - \\vec{w} = (\\vec{v}_x - \\vec{w}_x) + (\\vec{v}_y - \\vec{w}_y)`}\n

\n

\n Produce il vettore che parte da w e arriva a v.\n

\n \n \n

\n Prodotto scalare\n

\n

\n Si chiama scalare perchè il risultato è uno scalare, non un vettore.\n

\n

\n {r`\\vec{v} \\cdot \\vec{w} = \\left | \\vec{v} \\right | \\left | \\vec{w} \\right | \\cos \\alpha`}\n

\n

\n Produce il modulo della proiezione di {r`\\vec{a}`} su {r`\\vec{b}`}.\n

\n \n \n

\n Leggi di Newton\n

\n \n \n

\n 1ᵃ: Inerzia\n

\n

\n Se un corpo puntiforme ha forza risultante nulla, allora la sua velocità non cambia.\n

\n

\n {r`\\Sigma \\vec{F} = 0 \\Longleftrightarrow \\Delta v = 0`}\n

\n \n \n

\n 2ᵃ: Proporzionalità\n

\n

\n La forza risultante di un corpo è direttamente proporzionale alla sua accelerazione, e la costante di proporzionalità è la massa.\n

\n

\n {r`\\Sigma \\vec{F} = m \\vec{a}`}\n

\n \n \n

\n 3ᵃ: Azione e reazione\n

\n

\n Due corpi esercitano forze uguali e opposte uno sull'altro. \n

\n

\n {r`\\vec{F}_{21} = -\\vec{F}_{12}`}\n

\n \n \n

\n Forza di gravità\n

\n \n \n

\n Tra due corpi\n

\n

\n Due corpi puntiformi si attirano uno verso l'altro con forza:\n

\n

\n {r`\\left | \\vec{F} \\right | = G \\frac{m_1 m_2}{s^2}`}\n

\n

\n G è la costante di gravitazione universale e vale:\n

\n

\n {r`G = 6.67 \\cdot 10^{-11} \\frac{N m^2}{{kg}^2}`}\n

\n \n \n

\n Verso la Terra\n

\n

\n Se nel sistema di riferimento consideriamo la Terra ferma, allora un corpo è attratto verso la Terra con forza peso uguale a:\n

\n

\n {r`\\left | \\vec{F} \\right | = g m`}\n

\n

\n g è la costante di gravità della Terra, e vale:\n

\n

\n {r`g = 9.81 \\frac{m}{s^2}`}\n

\n \n \n

\n Su pianeti diversi\n

\n

\n Per pianeti diversi dalla Terra vale la stessa regola:\n

\n

\n {r`\\left | \\vec{F} \\right | = g m`}\n

\n

\n L'unica differenza è che cambia la costante di gravità:\n

\n

\n {r`g_{luna} = 1.62 \\frac{m}{s^2}`}\n

\n

\n {r`g_{marte} = 3.71 \\frac{m}{s^2}`}\n

\n \n \n

\n Forze di contatto\n

\n \n \n

\n Normale\n

\n

\n Si oppone alle forze applicate alla superficie di contatto.\n

\n

\n Un libro appoggiato su un tavolo ha la forza di gravità che lo attira verso il terreno e la forza normale che lo trattiene dal cadere. \n

\n \n \n

\n Attrito statico\n

\n

\n Impedisce a un corpo di muoversi se non viene spinto da una forza che supera una certa soglia:\n

\n

\n {r`\\left | \\vec{F} \\right | \\leq \\mu_{s} \\left | \\vec{F}_{normale} \\right |`}\n

\n \n \n

\n Attrito dinamico\n

\n

\n Rallenta i corpi che si stanno muovendo finchè essi non si fermano:\n

\n

\n {r`\\left | \\vec{F} \\right | \\leq \\mu_{d} \\left | \\vec{F}_{normale} \\right |`}\n

\n \n \n

\n Tensione\n

\n

\n E' forza trasmessa tra due estremi di una fune.\n

\n

\n Può essere redirezionata per mezzo di carrucole.\n

\n \n \n

\n Elastica\n

\n

\n Una molla cerca sempre di tornare alla sua posizione indeformata con forza:\n

\n

\n {r`F = -k x`}\n

\n

\n (E' negativa perchè la forza è opposta a quella applicata per deformarla.)\n

\n \n \n

\n Cinematica\n

\n \n \n

\n Spostamento\n

\n

\n È un vettore che indica la posizione di un corpo rispetto a un'origine.\n

\n

\n {r`\\Delta \\vec{s} = \\vec{s}(fine) - \\vec{s}(inizio)`}\n

\n \n \n

\n Velocità\n

\n

\n È un vettore che misura la variazione di posizione nel tempo.\n

\n

\n {r`\\vec{v} = \\frac{\\Delta \\vec{s}}{\\Delta t}`}\n

\n

\n Se si considera un intervallo di tempo infinitesimale si dice velocità istantanea:\n

\n

\n {r`\\vec{v} = \\lim_{\\Delta t \\to 0} \\frac{\\Delta \\vec{s}}{\\Delta t} = \\frac{d \\vec{s}}{dt}`}\n

\n \n \n

\n Accelerazione\n

\n

\n È un vettore che misura la variazione di velocità nel tempo.\n

\n

\n {r`\\vec{a} = \\frac{\\Delta \\vec{v}}{\\Delta t}`}\n

\n

\n Se si considera un intervallo di tempo infinitesimale si dice accelerazione istantanea:\n

\n

\n {r`\\vec{a} = \\lim_{\\Delta v \\to 0} \\frac{\\Delta \\vec{v}}{\\Delta t} = \\frac{d \\vec{v}}{d t} = \\frac{d^2 \\vec{s}}{d t^2}`}\n

\n \n \n

\n Quantità di moto (momento lineare)\n

\n

\n La quantità di moto è una proprietà vettoriale dei corpi:\n

\n

\n {r`\\vec{p} = m \\vec{v}`}\n

\n

\n Se la forza risultante è nulla, la quantità di moto non cambia.\n

\n

\n {r`\\Sigma \\vec{F} = 0 \\Longleftrightarrow \\Delta \\vec{p} = 0`}\n

\n \n \n

\n Moto rettilineo uniforme\n

\n \n \n

\n Spostamento\n

\n

\n La legge oraria è:\n

\n

\n {r`s(t) = v \\cdot \\Delta t + s(0)`}\n

\n \n \n

\n Velocità\n

\n

\n È costante:\n

\n

\n {r`v(t) = k`}\n

\n \n \n

\n Accelerazione\n

\n

\n La velocità non varia:\n

\n

\n {r`a(t) = 0`}\n

\n \n \n

\n Forze\n

\n

\n Si applica la prima legge di Newton:\n

\n

\n f(t) = 0\n

\n \n \n

\n Moto rettilineo uniformemente accelerato\n

\n \n \n

\n Spostamento\n

\n

\n La legge oraria è:\n

\n

\n {r`s(t) = \\frac{1}{2} a \\cdot (\\Delta t)^2 + v(0) \\cdot (\\Delta t) + s(0)`}\n

\n \n \n

\n Velocità\n

\n

\n È una retta:\n

\n

\n {r`v(t) = a \\Delta t + v(0)`}\n

\n \n \n

\n Accelerazione\n

\n

\n È costante:\n

\n

\n {r`a(t) = k`}\n

\n \n \n

\n Forze\n

\n

\n Si applica la prima legge di Newton:\n

\n

\n f(t) = m a\n

\n \n \n

\n Moto armonico semplice\n

\n \n \n

\n Ampiezza\n

\n

\n E' la distanza dal centro massima che raggiunge il corpo.\n

\n

\n (L'ampiezza di una sinusoide.)\n

\n \n \n

\n Velocità angolare\n

\n

\n Indica quanto in fretta cambia la posizione del corpo. \n

\n

\n Dipende dal periodo:\n

\n

\n {r`\\omega = \\frac{2 \\pi}{T}`}\n

\n \n \n

\n Spostamento\n

\n

\n E' una sinusoide:\n

\n

\n {r`s(t) = A \\sin (\\omega \\cdot t + \\phi)`}\n

\n \n \n

\n Velocità\n

\n

\n E' la sinusoide dello spostamento, sfasata di {r`\\frac{\\pi}{2}`}:\n

\n

\n {r`s(t) = A \\sin (\\omega \\cdot t + \\phi + \\frac{\\pi}{2})`}\n

\n \n \n

\n Accelerazione\n

\n

\n E' la sinusoide della velocità, sfasata di {r`\\frac{\\pi}{2}`}:\n

\n

\n {r`s(t) = A \\sin (\\omega \\cdot t + \\phi + \\pi)`}\n

\n \n \n

\n Forze\n

\n

\n Si applica la prima legge di Newton:\n

\n

\n f(t) = m a\n

\n \n \n

\n Moti composti\n

\n \n \n

\n Moto parabolico\n

\n

\n Il moto parabolico è dato sommando un moto rettilineo uniforme sull'asse orizzontale e un moto rettilineo uniformemente accelerato sull'asse verticale.\n

\n \n \n

\n Moto circolare uniforme\n

\n

\n Il moto parabolico è dato sommando due moti armonici semplici: uno sull'asse X, e l'altro, sfasato di {r`\\frac{\\pi}{2}`}, sull'asse Y.\n

\n \n \n

\n Moto circolare uniforme\n

\n \n \n

\n Velocità angolare\n

\n

\n Quanto cambia la fase nel tempo.\n

\n

\n {r`\\omega = \\frac{2 \\pi}{T}`}\n

\n \n \n

\n Fase\n

\n

\n E' l'angolo percorso dal corpo rispetto alla posizione iniziale.\n

\n

\n Si indica con {r`\\phi`}, e generalmente si usa in radianti.\n

\n \n \n

\n Velocità\n

\n

\n Si applicano le formule per la circonferenza:\n

\n

\n {r`v = \\frac{\\Delta s}{t} = \\frac{2 \\pi \\cdot r}{T}`}\n

\n \n \n

\n Accelerazione\n

\n

\n Il corpo ha sempre un accelerazione verso il centro che gli impedisce di abbandonare il moto: \n

\n

\n {r`a = \\frac{v^2}{r} = r \\cdot \\omega^2 = v \\cdot \\omega`}\n

\n \n \n

\n Forza centripeta\n

\n

\n È verso il centro e si calcola con:\n

\n

\n {r`F = m \\cdot a`}\n

\n \n \n

\n Lavoro ed energia\n

\n \n \n

\n Lavoro\n

\n

\n E' compiuto da una forza che sposta un corpo.\n

\n

\n {r`W = \\vec{F} \\cdot \\vec{s} = F \\cdot \\Delta s \\cdot cos(\\alpha )`}\n

\n

\n (Se la forza non è parallela allo spostamento, il prodotto scalare ci fa considerare solo la componente parallela.)\n

\n \n \n

\n Energia cinetica\n

\n

\n Un corpo ha energia cinetica in ogni momento uguale a:\n

\n

\n {r`E_c = \\frac{1}{2} m v^2`}\n

\n

\n Se una forza effettua lavoro su un corpo, cambia la sua energia cinetica pari al lavoro effettuato:\n

\n

\n {r`\\Delta E_c = W`}\n

\n \n \n

\n Energia potenziale gravitazionale\n

\n

\n Un corpo ha energia potenziale in ogni momento pari a: \n

\n

\n {r`E_{p_g} = m \\cdot g \\cdot h`}\n

\n

\n (Con h uguale a un altezza scelta come punto di riferimento.)\n

\n \n \n

\n Energia potenziale elastica\n

\n

\n Una molla ha sempre energia potenziale elastica pari a:\n

\n

\n {r`E_{p_e} = \\frac{1}{2} k x^2`}\n

\n \n \n

\n Forze conservative\n

\n

\n Sono conservative le forze per le quali il lavoro compiuto non dipende dal percorso seguito per andare dalla partenza all'arrivo.\n

\n

\n Ad esempio, è conservativa la forza di gravità, ma non è conservativa la forza di attrito.\n

\n

\n Se in un sistema ci sono solo forze conservative, allora l'energia meccanica totale si conserva:\n

\n

\n {r`E = E_k + E_p`}\n

\n \n \n

\n Potenza\n

\n

\n È la velocità di trasferimento di energia:\n

\n

\n {r`P = \\frac{\\Delta E}{\\Delta t}`}\n

\n \n \n
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"https://registry.npmjs.org/y18n/-/y18n-3.2.1.tgz", + "integrity": "sha1-bRX7qITAhnnA136I53WegR4H+kE=", + "dev": true + } + } + }, + "yargs-parser": { + "version": "7.0.0", + "resolved": "https://registry.npmjs.org/yargs-parser/-/yargs-parser-7.0.0.tgz", + "integrity": "sha1-jQrELxbqVd69MyyvTEA4s+P139k=", + "dev": true, + "requires": { + "camelcase": "^4.1.0" + }, + "dependencies": { + "camelcase": { + "version": "4.1.0", + "resolved": "https://registry.npmjs.org/camelcase/-/camelcase-4.1.0.tgz", + "integrity": "sha1-1UVjW+HjPFQmScaRc+Xeas+uNN0=", + "dev": true + } + } + }, + "yauzl": { + "version": "2.10.0", + "resolved": "https://registry.npmjs.org/yauzl/-/yauzl-2.10.0.tgz", + "integrity": "sha1-x+sXyT4RLLEIb6bY5R+wZnt5pfk=", + "dev": true, + "requires": { + "buffer-crc32": "~0.2.3", + "fd-slicer": "~1.1.0" + } + } + } +} diff --git a/package.json b/package.json index 71b4189..ba29606 100644 --- a/package.json +++ b/package.json @@ -1,7 +1,7 @@ { "private": true, "name": "appuntiweb", - "version": "0.0.0", + "version": "0.1.0", "license": "MIT", "scripts": { "start": "if-env NODE_ENV=production && npm run -s serve || npm run -s dev", @@ -27,4 +27,4 @@ "preact-compat": "^3.17.0", "preact-render-to-string": "^4.1.0" } -} \ No newline at end of file +} diff --git a/src/components/copyright.css b/src/components/copyright.css new file mode 100644 index 0000000..10d0e73 --- /dev/null +++ b/src/components/copyright.css @@ -0,0 +1,6 @@ +.copyright { + margin-top: 8px; + color: white; + text-align: center; + font-size: x-small; +} \ No newline at end of file diff --git a/src/components/copyright.js b/src/components/copyright.js new file mode 100644 index 0000000..f8b8913 --- /dev/null +++ b/src/components/copyright.js @@ -0,0 +1,8 @@ +import style from "./copyright.css"; +import { Component } from 'preact'; + +export default class Copyright extends Component { + render() { + return
© 2019 - Stefano Pigozzi - CC BY-SA 4.0
; + } +} \ No newline at end of file diff --git a/src/components/panel.css b/src/components/panel.css index 2766b6d..429cff8 100644 --- a/src/components/panel.css +++ b/src/components/panel.css @@ -4,4 +4,5 @@ border-radius: 4px; margin: 4px; height: calc(100% - 8px); + min-width: 200px; } \ No newline at end of file diff --git a/src/components/split.js b/src/components/split.js index f628ed7..71c9748 100644 --- a/src/components/split.js +++ b/src/components/split.js @@ -4,9 +4,15 @@ import { Component } from 'preact'; export default class Split extends Component { render() { let percent = 100 / this.props.children.count; - let children = this.props.children.map(element => { - return (
{element}
); - }); + let children = null; + if(Array.isArray(this.props.children)) { + children = this.props.children.map(element => { + return (
{element}
); + }); + } + else { + children =
{this.props.children}
; + } return
{children}
; } } \ No newline at end of file diff --git a/src/index.css b/src/index.css index 8c683e3..4dd9ecd 100644 --- a/src/index.css +++ b/src/index.css @@ -15,3 +15,16 @@ h1, h2, h3, h4, h5, h6 { color: #ffffff; font-weight: normal; } + +a { + color: #00caca; + text-decoration: none; +} + +a:hover { + color: #4affff; +} + +a:active { + color: #ffffff; +} \ No newline at end of file diff --git a/src/index.js b/src/index.js index fc652cc..79700fa 100644 --- a/src/index.js +++ b/src/index.js @@ -1,11 +1,15 @@ import './index.css'; import { Component } from 'preact'; -import Fisica from './pages/fisica' +import Fisica from './pages/fisica'; +import Copyright from './components/copyright'; export default class App extends Component { render() { return ( - +
+ + +
); } } diff --git a/src/pages/fisica.js b/src/pages/fisica.js index 4b8519d..2828d5b 100644 --- a/src/pages/fisica.js +++ b/src/pages/fisica.js @@ -4,6 +4,8 @@ import Latex from '../components/latex'; import Panel from '../components/panel'; import Split from '../components/split'; +const r = String.raw; + export default class Fisica extends Component { render() { return ( @@ -19,13 +21,13 @@ export default class Fisica extends Component { Usa le regole base della trigonometria:

- {String.raw`\vec{v} = \vec{v}_x + \vec{v}_y`} + {r`\vec{v} = \vec{v}_x + \vec{v}_y`}

- {String.raw`\left | \vec{v}_x \right | = \left | \vec{v} \right | \sin \alpha`} + {r`\left | \vec{v}_x \right | = \left | \vec{v} \right | \sin \alpha`}

- {String.raw`\left | \vec{v}_y \right | = \left | \vec{v} \right | \cos \alpha`} + {r`\left | \vec{v}_y \right | = \left | \vec{v} \right | \cos \alpha`}

@@ -36,7 +38,7 @@ export default class Fisica extends Component { Scomponi in componenti, poi sommali:

- {String.raw`\vec{v} + \vec{w} = (\vec{v}_x + \vec{w}_x) + (\vec{v}_y + \vec{w}_y)`} + {r`\vec{v} + \vec{w} = (\vec{v}_x + \vec{w}_x) + (\vec{v}_y + \vec{w}_y)`}

Produce il vettore risultante dall'applicazione della regola del parallelogramma. @@ -50,7 +52,7 @@ export default class Fisica extends Component { Alla fine è sempre una somma:

- {String.raw`\vec{v} - \vec{w} = (\vec{v}_x - \vec{w}_x) + (\vec{v}_y - \vec{w}_y)`} + {r`\vec{v} - \vec{w} = (\vec{v}_x - \vec{w}_x) + (\vec{v}_y - \vec{w}_y)`}

Produce il vettore che parte da w e arriva a v. @@ -64,10 +66,10 @@ export default class Fisica extends Component { Si chiama scalare perchè il risultato è uno scalare, non un vettore.

- {String.raw`\vec{v} \cdot \vec{w} = \left | \vec{v} \right | \left | \vec{w} \right | \cos \alpha`} + {r`\vec{v} \cdot \vec{w} = \left | \vec{v} \right | \left | \vec{w} \right | \cos \alpha`}

- Produce il modulo della proiezione di {String.raw`\vec{a}`} su {String.raw`\vec{b}`}. + Produce il modulo della proiezione di {r`\vec{a}`} su {r`\vec{b}`}.

@@ -83,7 +85,7 @@ export default class Fisica extends Component { Se un corpo puntiforme ha forza risultante nulla, allora la sua velocità non cambia.

- {String.raw`\Sigma \vec{F} = 0 \Longleftrightarrow \Delta v = 0`} + {r`\Sigma \vec{F} = 0 \Longleftrightarrow \Delta v = 0`}

@@ -94,7 +96,7 @@ export default class Fisica extends Component { La forza risultante di un corpo è direttamente proporzionale alla sua accelerazione, e la costante di proporzionalità è la massa.

- {String.raw`\Sigma \vec{F} = m \vec{a}`} + {r`\Sigma \vec{F} = m \vec{a}`}

@@ -105,7 +107,7 @@ export default class Fisica extends Component { Due corpi esercitano forze uguali e opposte uno sull'altro.

- {String.raw`\vec{F}_{21} = -\vec{F}_{12}`} + {r`\vec{F}_{21} = -\vec{F}_{12}`}

@@ -121,13 +123,13 @@ export default class Fisica extends Component { Due corpi puntiformi si attirano uno verso l'altro con forza:

- {String.raw`\left | \vec{F} \right | = G \frac{m_1 m_2}{s^2}`} + {r`\left | \vec{F} \right | = G \frac{m_1 m_2}{s^2}`}

G è la costante di gravitazione universale e vale:

- {String.raw`G = 6.67 \cdot 10^{-11} \frac{N m^2}{{kg}^2}`} + {r`G = 6.67 \cdot 10^{-11} \frac{N m^2}{{kg}^2}`}

@@ -138,13 +140,13 @@ export default class Fisica extends Component { Se nel sistema di riferimento consideriamo la Terra ferma, allora un corpo è attratto verso la Terra con forza peso uguale a:

- {String.raw`\left | \vec{F} \right | = g m`} + {r`\left | \vec{F} \right | = g m`}

g è la costante di gravità della Terra, e vale:

- {String.raw`g = 9.81 \frac{m}{s^2}`} + {r`g = 9.81 \frac{m}{s^2}`}

@@ -155,16 +157,16 @@ export default class Fisica extends Component { Per pianeti diversi dalla Terra vale la stessa regola:

- {String.raw`\left | \vec{F} \right | = g m`} + {r`\left | \vec{F} \right | = g m`}

L'unica differenza è che cambia la costante di gravità:

- {String.raw`g_{luna} = 1.62 \frac{m}{s^2}`} + {r`g_{luna} = 1.62 \frac{m}{s^2}`}

- {String.raw`g_{marte} = 3.71 \frac{m}{s^2}`} + {r`g_{marte} = 3.71 \frac{m}{s^2}`}

@@ -191,7 +193,7 @@ export default class Fisica extends Component { Impedisce a un corpo di muoversi se non viene spinto da una forza che supera una certa soglia:

- {String.raw`\left | \vec{F} \right | \leq \mu_{s} \left | \vec{F}_{normale} \right |`} + {r`\left | \vec{F} \right | \leq \mu_{s} \left | \vec{F}_{normale} \right |`}

@@ -202,7 +204,7 @@ export default class Fisica extends Component { Rallenta i corpi che si stanno muovendo finchè essi non si fermano:

- {String.raw`\left | \vec{F} \right | \leq \mu_{d} \left | \vec{F}_{normale} \right |`} + {r`\left | \vec{F} \right | \leq \mu_{d} \left | \vec{F}_{normale} \right |`}

@@ -224,7 +226,7 @@ export default class Fisica extends Component { Una molla cerca sempre di tornare alla sua posizione indeformata con forza:

- {String.raw`F = -k x`} + {r`F = -k x`}

(E' negativa perchè la forza è opposta a quella applicata per deformarla.) @@ -243,7 +245,7 @@ export default class Fisica extends Component { È un vettore che indica la posizione di un corpo rispetto a un'origine.

- {String.raw`\Delta \vec{s} = \vec{s}(fine) - \vec{s}(inizio)`} + {r`\Delta \vec{s} = \vec{s}(fine) - \vec{s}(inizio)`}

@@ -254,13 +256,13 @@ export default class Fisica extends Component { È un vettore che misura la variazione di posizione nel tempo.

- {String.raw`\vec{v} = \frac{\Delta \vec{s}}{\Delta t}`} + {r`\vec{v} = \frac{\Delta \vec{s}}{\Delta t}`}

Se si considera un intervallo di tempo infinitesimale si dice velocità istantanea:

- {String.raw`\vec{v} = \lim_{\Delta t \to 0} \frac{\Delta \vec{s}}{\Delta t} = \frac{d \vec{s}}{dt}`} + {r`\vec{v} = \lim_{\Delta t \to 0} \frac{\Delta \vec{s}}{\Delta t} = \frac{d \vec{s}}{dt}`}

@@ -271,13 +273,13 @@ export default class Fisica extends Component { È un vettore che misura la variazione di velocità nel tempo.

- {String.raw`\vec{a} = \frac{\Delta \vec{v}}{\Delta t}`} + {r`\vec{a} = \frac{\Delta \vec{v}}{\Delta t}`}

Se si considera un intervallo di tempo infinitesimale si dice accelerazione istantanea:

- {String.raw`\vec{a} = \lim_{\Delta v \to 0} \frac{\Delta \vec{v}}{\Delta t} = \frac{d \vec{v}}{d t} = \frac{d^2 \vec{s}}{d t^2}`} + {r`\vec{a} = \lim_{\Delta v \to 0} \frac{\Delta \vec{v}}{\Delta t} = \frac{d \vec{v}}{d t} = \frac{d^2 \vec{s}}{d t^2}`}

@@ -288,13 +290,13 @@ export default class Fisica extends Component { La quantità di moto è una proprietà vettoriale dei corpi:

- {String.raw`\vec{p} = m \vec{v}`} + {r`\vec{p} = m \vec{v}`}

Se la forza risultante è nulla, la quantità di moto non cambia.

- {String.raw`\Sigma \vec{F} = 0 \Longleftrightarrow \Delta \vec{p} = 0`} + {r`\Sigma \vec{F} = 0 \Longleftrightarrow \Delta \vec{p} = 0`}

@@ -310,7 +312,7 @@ export default class Fisica extends Component { La legge oraria è:

- {String.raw`s(t) = v \cdot \Delta t + s(0)`} + {r`s(t) = v \cdot \Delta t + s(0)`}

@@ -321,7 +323,7 @@ export default class Fisica extends Component { È costante:

- {String.raw`v(t) = k`} + {r`v(t) = k`}

@@ -332,7 +334,7 @@ export default class Fisica extends Component { La velocità non varia:

- {String.raw`a(t) = 0`} + {r`a(t) = 0`}

@@ -359,7 +361,7 @@ export default class Fisica extends Component { La legge oraria è:

- {String.raw`s(t) = \frac{1}{2} a \cdot (\Delta t)^2 + v(0) \cdot (\Delta t) + s(0)`} + {r`s(t) = \frac{1}{2} a \cdot (\Delta t)^2 + v(0) \cdot (\Delta t) + s(0)`}

@@ -370,7 +372,7 @@ export default class Fisica extends Component { È una retta:

- {String.raw`v(t) = a \Delta t + v(0)`} + {r`v(t) = a \Delta t + v(0)`}

@@ -381,7 +383,7 @@ export default class Fisica extends Component { È costante:

- {String.raw`a(t) = k`} + {r`a(t) = k`}

@@ -422,7 +424,7 @@ export default class Fisica extends Component { Dipende dal periodo:

- {String.raw`\omega = \frac{2 \pi}{T}`} + {r`\omega = \frac{2 \pi}{T}`}

@@ -433,7 +435,7 @@ export default class Fisica extends Component { E' una sinusoide:

- {String.raw`s(t) = A \sin (\omega \cdot t + \phi)`} + {r`s(t) = A \sin (\omega \cdot t + \phi)`}

@@ -441,10 +443,10 @@ export default class Fisica extends Component { Velocità

- E' la sinusoide dello spostamento, sfasata di {String.raw`\frac{\pi}{2}`}: + E' la sinusoide dello spostamento, sfasata di {r`\frac{\pi}{2}`}:

- {String.raw`s(t) = A \sin (\omega \cdot t + \phi + \frac{\pi}{2})`} + {r`s(t) = A \sin (\omega \cdot t + \phi + \frac{\pi}{2})`}

@@ -452,10 +454,10 @@ export default class Fisica extends Component { Accelerazione

- E' la sinusoide della velocità, sfasata di {String.raw`\frac{\pi}{2}`}: + E' la sinusoide della velocità, sfasata di {r`\frac{\pi}{2}`}:

- {String.raw`s(t) = A \sin (\omega \cdot t + \phi + \pi)`} + {r`s(t) = A \sin (\omega \cdot t + \phi + \pi)`}

@@ -487,11 +489,160 @@ export default class Fisica extends Component { Moto circolare uniforme

- Il moto parabolico è dato sommando due moti armonici semplici: uno sull'asse X, e l'altro, sfasato di {String.raw`\frac{\pi}{2}`}, sull'asse Y. + Il moto parabolico è dato sommando due moti armonici semplici: uno sull'asse X, e l'altro, sfasato di {r`\frac{\pi}{2}`}, sull'asse Y. +

+ + +

+ Moto circolare uniforme +

+ + +

+ Velocità angolare +

+

+ Quanto cambia la fase nel tempo. +

+

+ {r`\omega = \frac{2 \pi}{T}`} +

+ + +

+ Fase +

+

+ E' l'angolo percorso dal corpo rispetto alla posizione iniziale. +

+

+ Si indica con {r`\phi`}, e generalmente si usa in radianti. +

+ + +

+ Velocità +

+

+ Si applicano le formule per la circonferenza: +

+

+ {r`v = \frac{\Delta s}{t} = \frac{2 \pi \cdot r}{T}`} +

+ + +

+ Accelerazione +

+

+ Il corpo ha sempre un accelerazione verso il centro che gli impedisce di abbandonare il moto: +

+

+ {r`a = \frac{v^2}{r} = r \cdot \omega^2 = v \cdot \omega`} +

+ + +

+ Forza centripeta +

+

+ È verso il centro e si calcola con: +

+

+ {r`F = m \cdot a`} +

+ + +

+ Lavoro ed energia +

+ + +

+ Lavoro +

+

+ E' compiuto da una forza che sposta un corpo. +

+

+ {r`W = \vec{F} \cdot \vec{s} = F \cdot \Delta s \cdot cos(\alpha )`} +

+

+ (Se la forza non è parallela allo spostamento, il prodotto scalare ci fa considerare solo la componente parallela.) +

+ + +

+ Energia cinetica +

+

+ Un corpo ha energia cinetica in ogni momento uguale a: +

+

+ {r`E_c = \frac{1}{2} m v^2`} +

+

+ Se una forza effettua lavoro su un corpo, cambia la sua energia cinetica pari al lavoro effettuato: +

+

+ {r`\Delta E_c = W`} +

+ + +

+ Energia potenziale gravitazionale +

+

+ Un corpo ha energia potenziale in ogni momento pari a: +

+

+ {r`E_{p_g} = m \cdot g \cdot h`} +

+

+ (Con h uguale a un altezza scelta come punto di riferimento.) +

+ + +

+ Energia potenziale elastica +

+

+ Una molla ha sempre energia potenziale elastica pari a: +

+

+ {r`E_{p_e} = \frac{1}{2} k x^2`} +

+ + +

+ Forze conservative +

+

+ Sono conservative le forze per le quali il lavoro compiuto non dipende dal percorso seguito per andare dalla partenza all'arrivo. +

+

+ Ad esempio, è conservativa la forza di gravità, ma non è conservativa la forza di attrito. +

+

+ Se in un sistema ci sono solo forze conservative, allora l'energia meccanica totale si conserva: +

+

+ {r`E = E_k + E_p`} +

+ + +

+ Potenza +

+

+ È la velocità di trasferimento di energia: +

+

+ {r`P = \frac{\Delta E}{\Delta t}`}

) } -} \ No newline at end of file +}