I = ∑ M_{i}R_{i}^{2}

I_{central} = \frac{1}{12}mL^{2}

I_{shifted} = I_{central} + mx^{2}

= \frac{1}{12}mL^{2} + mx^{2}

I_{2−rod} = I_{central} + I_{shifted}

= (\frac{1}{12}mL^{2}) + (\frac{1}{12}mL^{2} + Mx^{2})

surface density σ = \frac{M}{A} = \frac{M}{L^{2}} dM = σ ∗ ( area ) = σ ∗ (L ∗ dx)

\begin{aligned}dI & = (\frac{1}{12} dM L^{2}) + (dM x^{2}) \\ dI & = (\frac{1}{12}(σLdx)L^{2}) + ((σLdx)x^{2}) \\ dI & = (\frac{1}{12}σL^{3}dx) + (σLx^{2}dx) \end{aligned}

I = ∑ m_{i} R_{i}^{2} mass of each little piece = dm

I = ∫ dm R^{2}

\begin{aligned}W & = ∫ F⃗ · dx⃗ \\ W & = \frac{1}{2}mv_{f}^{2} − \frac{1}{2}mv_{i}^{2} \\ P & = F⃗ · v⃗ \end{aligned}  \begin{aligned}W & = ∫ τ⃗ · dθ⃗ \\ W & = \frac{1}{2}Iω_{f}^{2} − \frac{1}{2}Iω_{i}^{2} \\ P & = τ⃗ · ω⃗ \end{aligned}

F⃗ = m a⃗

τ⃗ = I α⃗

units of τ = (kg ∗ m^{2}) (\frac{rad}{s^{2}}) = (\frac{kg ∗ m}{s^{2}}) (m) = N ∗ m

|τ| = |r| |F | sin(θ)

I_{off−center} = I_{central} + Md^{2}

I_{central} = \frac{1}{12} M (L^{2} + L^{2}) = \frac{1}{6} M L^{2}

d = \frac{\sqrt{2}}{2} L

I_{off−center} = \frac{1}{6}ML^{2} + \frac{1}{2}ML^{2} = \frac{2}{3}ML^{2}

λ(x) = 2 \frac{kg}{m} − 0.5 \frac{kg}{m^{3}} x^{2}

dm = λ(x)dx = 2 \frac{kg}{m} dx − 0.5 \frac{kg}{m^{3}} x^{2}dx

dI = dm x^{2} = 2 \frac{kg}{m} x^{2}dx − 0.5 \frac{kg}{m^{3}} x^{4}dx

I = ∫ dI = ∫_{x=0}^{x=1 m} 2 \frac{kg}{m} x^{2}dx − ∫_{x=0}^{x=1 m} 0.5 \frac{kg}{m^{3}} x^{4}dx

I = 2 \frac{kg}{m} [\frac{1}{3} (1 m)^{3}] − 0.5 \frac{kg}{m^{3}} [\frac{1}{5} (1 m)^{5}]