KE = \frac{1}{2}M_{1}v_{1}^{2}

= \frac{1}{2}M_{1}(R_{1}ω)^{2}

= \frac{1}{2}M_{1}R_{1}^{2}ω^{2}

= \frac{1}{2}(M_{1}R_{1}^{2})ω^{2}

I = MR^{2}

KE = \frac{1}{2}I_{1}ω^{2}

KE = \frac{1}{2}M_{1}v_{1}^{2} + \frac{1}{2}M_{2}v_{2}^{2} + \frac{1}{2}M_{3}v_{3}^{2}

= \frac{1}{2}(M_{1}R_{1}^{2})ω^{2} + \frac{1}{2}(M_{2}R_{2}^{2})ω^{2} + \frac{1}{2}(M_{3}R_{3}^{2})ω^{2}

= \frac{1}{2}I_{1}ω^{2} + \frac{1}{2}I_{2}ω^{2} + \frac{1}{2}I_{3}ω^{2}

= \frac{1}{2}(I_{1} + I_{2} + I_{3})ω^{2}

= \frac{1}{2}I_{tot}ω^{2}

λ(x) = 3\frac{kg}{m} + (3\frac{kg}{m} )(\frac{x}{10m})

(KE − E_{US}) = \frac{1}{2} I ω_{new}^{2}

λ = \frac{M}{L}

dm = λ dx

dI = (mass)(distance from rotation axis)^{2} = dm x^{2}

I = ∫ dI = ∫_{x=0}^{x=L} dm x^{2} = ∫_{x=0}^{x=L} λ dx x^{2}

I = λ ∫_{0}^{L} x^{2}dx = λ (\frac{1}{3}L^{3}) = \frac{M}{L} \frac{1}{3} L^{3} = \frac{1}{3} ML^{2}

I = ∫ dI = ∫_{x=−L/2}^{x=L/2} dm x^{2} = ∫_{x=−L/2}^{x=L/2} λ dx x^{2}

I = λ (\frac{1}{3} \frac{L^{3}}{8} − [−\frac{1}{3} \frac{L^{3}}{8} ]) = \frac{M}{L} (\frac{2}{24} L^{3}) = \frac{1}{12} ML^{2}

λ(x) = A + Bx

dm = λ dx = A dx + B xdx

dI = dm x^{2} = (A dx) x^{2} + (B xdx) x^{2}

I = ∫ dI = ∫_{x=0}^{x=L} (A x^{2}dx) + ∫_{x=0}^{x=L} (B x^{3}dx)

I = \frac{1}{3} AL^{3} + \frac{1}{4} BL^{4}