politja horitzontal y molla horitzontal

m_{1}·d_{tt}^{2}[x_{1}] = T+(-k)·x_{1}
m_{2}·d_{tt}^{2}[x_{2}] = F+(-T)


( m_{1}+m_{2} )·d_{tt}^{2}[x] = F+(-k)·x


x(t) = cos( ( k/(m_{1}+m_{2}) )^{(1/2)}·t )+i·sin( ( k/(m_{1}+m_{2}) )^{(1/2)}·t )+( F/k )


T = F+m_{2}·( cos( ( k/(m_{1}+m_{2}) )^{(1/2)}·t )+i·sin( ( k/(m_{1}+m_{2}) )^{(1/2)}·t ) )

sistemes no inercials

d_{t}[x] = d_{t}[y]+(-1)·R·d_{t}[s(t)]


d_{tt}^{2}[x] = d_{tt}^{2}[y]+(-1)·R·d_{tt}^{2}[s(t)]


d_{tt}^{2}[x] = d_{tt}^{2}[y]+(-1)·( R/s(t) )·d_{t}[s(t)]^{2}


d_{t}[x] = d_{t}[y]+(-1)·R·ln(s(t)) [o(t)o] s(t)

índex de física

índex de etiquetes de física

politja doble fixada en un extrem en un sostre y estirada per l'altre

( m or q ) esta en la politja central no extrem de la corda.


politja doble amb una força constant en el extrem de la corda.
m·d_{tt}^{2}[y(t)] = q·g+(-1)·( F/2 )


politja doble amb una molla en el extrem de la corda.
m·d_{tt}^{2}[y(t)] = q·g+(-1)·( (k/2)·y(t) )


politja doble amb una força dependent del temps en el extrem de la corda.
m·d_{tt}^{2}[y(t)] = q·g+(-1)·( F(t)/2 )


politja doble amb una càrrega en el extrem de la corda.
m·d_{tt}^{2}[y(t)] = q·g+(-1)·( T/2 )
m_{1}·d_{tt}^{2}[y_{1}(t)] = (-1)·q_{1}·g+T


(m_{1}+2m)·d_{tt}^{2}[y_{1}(t)] = 2·q·g+(-1)·q_{1}·g


T = ( (m_{1}(2q)+(2m)q_{1})/(m_{1}+2m) )·g


Si m_{1} = 0 ==> T = q_{1}·g


(-T) = 2m·( (2·q·g+(-1)·q_{1}·g)/(m_{1}+2m) )+( (-1)·(m_{1}+2m)·2qg/(m_{1}+2m) )
T = m_{1}·( (2·q·g+(-1)·q_{1}·g)/(m_{1}+2m) )+( (m_{1}+2m)·q_{1}g/(m_{1}+2m) )

politja simple

m_{1}·d_{tt}^{2}[y_{1}(t)] = ( q_{1}·g+(-1)·T )
m_{2}·d_{tt}^{2}[y_{2}(t)] = ( (-1)·q_{2}·g+T )


(m_{1}+m_{2})·d_{tt}^{2}[y_{1}(t)] = ( q_{1}+(-1)q_{2} )·g


T = ( (m_{2}·q_{1}+m_{1}·q_{2})/(m_{1}+m_{2}) )·g


(-T) = m_{1}·( ( q_{1}+(-1)q_{2} )/(m_{1}+m_{2}) )·g+...
...( (-1)·((m_{1}+m_{2})·q_{1})/(m_{1}+m_{2}) )·g


T = m_{2}·( ( q_{1}+(-1)q_{2} )/(m_{1}+m_{2}) )·g+...
...( ((m_{1}+m_{2})·q_{2})/(m_{1}+m_{2}) )·g


politja simple sense càrrega  estirada per una força constant.
m·d_{tt}^{2}[y(t)] = ( q_{1}·g+q_{2}·g )+(-1)·F


politja simple sense càrrega penjada de una molla.
m·d_{tt}^{2}[y(t)] = ( q_{1}·g+q_{2}·g )+(-1)·k·y(t)


politja simple sense càrrega estirada per una força dependent del temps.
m·d_{tt}^{2}[y(t)] = ( q_{1}·g+q_{2}·g )+(-1)·F(t)

politja triple

( m or q ) esta en la politja central.


politja triple amb dos forçes constants en els extrems de la corda.
m·d_{tt}^{2}[y(t)] = q·g+(-1)·( F_{1}+F_{2} )


politja triple amb dos molles en els extrems de la corda.
m·d_{tt}^{2}[y(t)] = q·g+(-1)·( k_{1}·y(t)+k_{2}·y(t) )


politja triple amb dos forçes dependents del temps en els extrems de la corda.
m·d_{tt}^{2}[y(t)] = q·g+(-1)·( F_{1}(t)+F_{2}(t) )


politja triple amb dos càrregues en els extrems de la corda.
m·d_{tt}^{2}[y(t)] = q·g+(-1)·( T_{1}+T_{2} )
m_{1}·d_{tt}^{2}[y_{1}(t)] = (-1)·q_{1}·g+T_{1}
m_{2}·d_{tt}^{2}[y_{2}(t)] = (-1)·q_{2}·g+T_{2}


(m_{1}+m·( (n+(-k))/n ))·d_{tt}^{2}[y_{1}(t)] = q·( (n+(-k))/n )·g+(-1)·q_{1}·g
(m_{2}+m·( k/n ))·d_{tt}^{2}[y_{2}(t)] = q·( k/n )·g+(-1)·q_{2}·g


T_{1} = ( (m_{1}q·( (n+(-k))/n )+q_{1}m·( (n+(-k))/n ))/(m_{1}+m( (n+(-k))/n) ) )·g
T_{2} = ( (m_{2}q·( k/n )+q_{2}m·( k/n ))/(m_{2}+m( k/n )) )·g


si m_{1}=0 ==> T_{1}=q_{1}g
si m_{2}=0 ==> T_{2}=q_{2}g

mecànica de colisió de una partícula amb un extrem de una barra

txoc de una partícula en un extrem de una barra a velocitat constant V:
m_{1} = k·m_{2}


m_{1}·d_{t}[x(t_{0})] = m_{1}·d_{t}[x(t_{1})] + m_{2}·d_{t}[s(t_{2})]·R


m_{1}·V = m_{1}·d_{t}[x(t_{1})] + m_{2}·d_{t}[s(t_{2})]·R


txoc inelástic:
si d_{t}[x(t_{1})] = d_{t}[s(t_{2})]·R ==>


d_{t}[x(t_{1})] = V·( m_{1}/(m_{1}+m_{2}) )


d_{t}[x(t_{1})] = V·( k/(k+1) )
d_{t}[s(t_{2})] = (V/R)·(k/(k+1) )


txoc elástic


m_{1}·V = m_{1}·(-V) + m_{2}·d_{t}[s(t_{2})]·R


d_{t}[s(t_{2})] =  (V/R)·(m_{1}/m_{2})
d_{t}[s(t_{2})] =  (V/R)·k

mecànica el trampolín parabólic

qgH + (-1)·qgy + (-1)·(m/2)·d_{t}[y]^{2} = qgy + (m/2)·d_{t}[y]^{2}


y(t) = (-1)·(1/2)·( (qg)/m )·t^{2} + (H/2)


d_{t}[y(t)] = (-1)·( (qg)/m )·t


d_{t}[y(t)] = 0 <==> y(t) = (H/2)


d_{t}[y(t)] = a <==> y(t) = (-1)·(1/2)·( m/(qg) )·a^{2} + (H/2)


0 = (-1)·(1/2)·( (qg)/m )·t^{2} + (H/2)


(H/2) = (1/2)·( (qg)/m )·t^{2}
t = ( ( (Hm)/(qg) ) )^{(1/2)}


qg(H/2) = (m/2)·d_{t}[x]^{2}


x(t) = ( ( (qgH)/m ) )^{(1/2)}·t


abast:
x(( ( (Hm)/(qg) ) )^{(1/2)}) = ( ( ( (qgH)/m ) )^{(1/2)} )·( ( ( (Hm)/(qg) ) )^{(1/2)} )
x(( ( (Hm)/(qg) ) )^{(1/2)}) = H


trayectoria de vuelo:
y = (-1)·( 1/(2H) )·x^{2}+(H/2)


y = (H/2) <==> x = 0


0 = (-1)·( 1/(2H) )·H^{2}+(H/2) <==> ( x = H or x = (-H) )


trayectoria de la rampa:
y = ( 1/(2H) )·x^{2}+(H/2)


y = (H/2) <==> x = 0


H = ( 1/(2H) )·H^{2}+(H/2) <==> ( x = H or x = (-H) )

mecànica clàssica arrel cúbica de la posició

m·d_{t}[x(t)] = a·( x(t) )^{(1/3)}


m·d_{tt}^{2}[x(t)] = (1/3)·a·( x(t) )^{(-1)(2/3)}·d_{t}[x(t)]


m·d_{tt}^{2}[x(t)] = (1/3)·(a^{2}/m)·( x(t) )^{(-1)(1/3)}


( x(t) )^{(1/3)}·d_{tt}^{2}[x(t)] = (1/3)·(a^{2}/m^{2})


d_{t}[x(t)]·d_{tt}^{2}[x(t)] = (1/3)·(a^{3}/m^{3})


(1/2)·d_{t}[x(t)]^{2} = (1/3)·(a^{3}/m^{3})·t


d_{t}[x(t)] = ( (2/3)·(a^{3}/m^{3}) )^{(1/2)}·t^{(1/2)}


x(t) = ( (2/3)·(a/m) )^{(3/2)}·t^{(3/2)}


d_{tt}^{2}[x(t)] = (1/2)·( (2/3)·(a^{3}/m^{3}) )^{(1/2)}·t^{(-1)(1/2)}


E(t) = ∫ [ (m/2)·( (2/3)·(a^{3}/m^{3}) )^{(1/2)}·t^{(-1)(1/2)}·( (2/3)·(a^{3}/m^{3}) )^{(1/2)}·t^{(1/2)}) ] d[t]
E(t) = ∫ [ (m/2)·( (2/3)·(a^{3}/m^{3}) ) ] d[t]


E(t) = (1/3)·(a^{3}/m^{2})·t


(m/2)·d_{t}[x(t)]^{2} = (1/3)·(a^{3}/m^{2})·t

mecànica clàssica arrel cuadrada de la posició

m·d_{t}[x(t)] = a·( x(t) )^{(1/2)}


m·d_{tt}^{2}[x(t)] = (1/2)·a·( x(t) )^{(-1)(1/2)}·d_{t}[x(t)]


m·d_{tt}^{2}[x(t)] = (1/2)·(a^{2}/m)


d_{tt}^{2}[x(t)] = (1/2)·(a^{2}/m^{2})


d_{t}[x(t)] = (1/2)·(a^{2}/m^{2})·t


x(t) = (1/4)·(a^{2}/m^{2})·t^{2}


E(t) = ∫ [ ( (1/2)·(a^{2}/m) )·( (1/2)·(a^{2}/m^{2})·t ) ] d[t]


E(t) = (1/8)·(a^{4}/m^{3})·t^{2}


(m/2)·d_{t}[x(t)]^{2} = (1/8)·(a^{4}/m^{3})·t^{2}

mecànica en un medi resistent

F_{x} = at^{n}


m·d_{tt}^{2}[x(t)] = F_{x}+(-1)·k·d_{t}[x(t)]


d_{tt}^{2}[x(t)] = (1/m)( at^{n}+(-1)·k·d_{t}[x(t)] )


d_{t}[x(t)] = (1/m)( ( a/(n+1) )t^{n+1}+(-1)·k·x(t) )


d_{t}[x(t)]+(k/m)·x(t) = (1/m)( ( a/(n+1) )t^{n+1} )


x(t) = e^{(-1)(k/m)t}·∫ [ (1/m)( a/(n+1) )t^{n+1}·e^{(k/m)t} ] d[t]

mecànica de una caisha estirada per una força vertical

F = at^{n}


m·d_{tt}^{2}[y(t)] = F+(-1)·qg


d_{tt}^{2}[y(t)] = (1/m)( at^{n}+(-1)·qg )


d_{t}[y(t)] = (1/m)( ( a/(n+1) )t^{n+1}+(-1)·qgt )


y(t) = (1/m)( ( a/((n+1)(n+2)) )t^{n+2}+(-1)·(1/2)·qg·t^{2} )
y(t) = (1/m)( ( a/((n+1)(n+2)) )t^{n}+(-1)·(1/2)·qg )·t^{2}


y(t) = 0 <==> ( t_{0} = 0 or t_{k} = ( (1/2)·qg·( ((n+1)(n+2))/a ) )^{(1/n)} )


d_{tt}^{2}[y(t_{k})] = (1/m)·qg·( (1/2)(n+1)(n+2)+(-1) )


d_{t}[y(t_{k})] = (1/m)·qg·( (1/2)(n+2)+(-1) )·( (1/2)·qg·( ((n+1)(n+2))/a ) )^{(1/n)}

mecànica clàssica

F = at^{n}


m·d_{tt}^{2}[x(t)] = at^{n}
d_{tt}^{2}[x(t)] = (a/m)·t^{n}


d_{t}[x(t)] = ( a/((n+1)m) )·t^{n+1}


x(t) = ( a/((n+1)(n+2)m) )·t^{n+2}


E(t) = ∫ [ at^{n} ] d[x]
E(t) = ∫ [ at^{n}·d_{t}[x] ] d[t]
E(t) = ∫ [ at^{n}·( a/((n+1)m) )·t^{n+1} ] d[t]


E(t) = ( a^{2}/(2(n+1)^{2}m) )·t^{2(n+1)}


(m/2)·d_{t}[x(t)]^{2} = ( a^{2}/(2(n+1)^{2}m) )·t^{2(n+1)}

mecànica clàssica

F = k·sin(at)


m·d_{tt}^{2}[x(t)] = k·sin(at)
d_{tt}^{2}[x(t)] = (k/m)·sin(at)


d_{t}[x(t)] = (-1)·( k/(am) )·cos(at)


x(t) = (-1)·( k/(a^{2}m) )·sin(at)


E(t) = ∫ [ k·sin(at) ] d[x]
E(t) = ∫ [ k·sin(at)·d_{t}[x] ] d[t]
E(t) = ∫ [ k·sin(at)·(-1)·( k/(am) )·cos(at) ] d[t]
E(t) = ∫ [ ( k^{2}/(am) )·cos(at)·(-1)·sin(at) ] d[t]


E(t) = ( k^{2}/(2a^{2}m) )·( cos(at) )^{2}


(m/2)·d_{t}[x(t)]^{2} = ( k^{2}/(2a^{2}m) )·( cos(at) )^{2}

mecànica clàssica

F = ke^{at}


m·d_{tt}^{2}[x(t)] = ke^{at}
d_{tt}^{2}[x(t)] = (k/m)·e^{at}


d_{t}[x(t)] = ( k/(am) )·e^{at}


x(t) = ( k/(a^{2}m) )·e^{at}


E(t) = ∫ [ ke^{at} ] d[x]
E(t) = ∫ [ ke^{at}·d_{t}[x] ] d[t]
E(t) = ∫ [ ke^{at}·( k/(am) )·e^{at} ] d[t]
E(t) = ∫ [ ( k^{2}/(am) )·e^{2at} ] d[t]


E(t) = ( k^{2}/(2a^{2}m) )·e^{2at}


(m/2)·d_{t}[x(t)]^{2} = ( k^{2}/(2a^{2}m) )·e^{2at}

mecànica clàssica

F = a


m·d_{tt}^{2}[y(t)] = a
d_{tt}^{2}[y(t)] = (a/m)


d_{t}[y(t)] = (a/m)·t


y(t) = (a/2m)·t^{2}


E(t) = ∫ [ a ] d[y]
E(t) = ∫ [ a·d_{t}[y] ] d[t]
E(t) = ∫ [ a·(a/m)·t ] d[t]


E(t) = (a^{2}/2m)·t^{2}


(m/2)·d_{t}[y(t)]^{2} = (a^{2}/2m)·t^{2}

mecànica clàssica

F = at


m·d_{tt}^{2}[x(t)] = at
d_{tt}^{2}[x(t)] = (a/m)·t


d_{t}[x(t)] = ( a/(2m) )·t^{2}


x(t) = ( a/(6m) )·t^{3}


E(t) = ∫ [ at ] d[x]
E(t) = ∫ [ at·d_{t}[x] ] d[t]
E(t) = ∫ [ at·( a/(2m) )·t^{2} ] d[t]


E(t) = ( a^{2}/(8m) )·t^{4}


(m/2)·d_{t}[x(t)]^{2} = ( a^{2}/(8m) )·t^{4}