Momentos De Inercia

Cálculos de momento de inercia

Para un disco:

I_cm=∫_0^M▒r^2 dm = ρ∫_0^M▒r^2 dV = M/(πR^2 h) ∫_0^R▒r^2 2πrhdr = 2M/R^2 ∫_0^R▒r^3 dr = 2M/R^2 (R^4/4) → I_cm= 1/2 MR^2

dm=ρdV

dV=2πrhdr

ρ=M/(πR^2 h)

Para una barra:

I_cm=∫_0^M▒〖r^2 dm = ρ∫_0^M▒r^2 dV =M/abc ∫_0^V▒(z^2+y^2 )dV =M/abc [∫_0^V▒z^2 dV+∫_0^V▒y^2 dV] 〗

I_cm=M/abc [∫_(-a/2)^(q/2)▒z^2 bcdz+∫_(-b/2)^(b/2)▒y^2 acdy ] =M/abc [bc(a^3/12)+ac(b^3/12)] → I_cm=(M(a^2+b^2))/12

dm=ρdV

ρ=M/abc

z dV=bc dz

y dV=ac dy

Para cilindro hueco:

I_cm=∫_0^M▒r^2 dm = ρ∫_0^M▒r^2 dV = 2ρπh∫_(R_1)^(R_2)▒r^2 rdr = 2ρπh∫_(R_1)^(R_2)▒r^3 dr

I_cm= 2πhρ ((〖R_2〗^4- 〖R_1〗^4)/4)=2πh M/(πh(〖R_1〗^2-〖R_2〗^2))*((〖R_2〗^2- 〖R_1〗^2)(〖R_2〗^2+ 〖R_1〗^2))/4 → I_cm= 1/2 M(〖R_2〗^2+ 〖R_1〗^2)

dm=ρdV

dV=2πrhdr

ρ=M/(V_exterior-V_interior )=M/(πh(〖R_1〗^2-〖R_2〗^2))

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