3C (Dynamics and Relativity)

The parallel axis theorem says that for an axis through

x

, parallel to the centre-of-mass axis in direction

\hat{d}

I_{T} (x) = I_{T} (x_{T}) + M_{T} ∣ (x - x_{T}) \times \hat{d} ∣^{2} .

Let

ρ_{A}^{2} = ∣ (x_{A} - x_{C}) \times \hat{d} ∣^{2}, ρ_{B}^{2} = ∣ (x_{B} - x_{C}) \times \hat{d} ∣^{2} .

Moments about the axis through

x_{C}

add, so

I_{C} (x_{C}) = I_{A} (x_{A}) + M_{A} ρ_{A}^{2} + I_{B} (x_{B}) + M_{B} ρ_{B}^{2} .

Using

M_{A} = M_{C} - M_{B}

and solving gives

M_{B} = \frac{I _{C} ( x _{C} ) - I _{A} ( x _{A} ) - I _{B} ( x _{B} ) - M _{C} ρ _{A}^{2}}{ρ _{B}^{2} - ρ _{A}^{2}}

when the denominator is nonzero. In the degenerate equal-distance case, this inertia equation alone does not determine

M_{B}

; the centre-of-mass equation

M_{C} x_{C} = M_{A} x_{A} + M_{B} x_{B}

supplies the remaining information.

b

For a lamina in the

x y

I_{x} = \int y^{2} d m, I_{y} = \int x^{2} d m, I_{z} = \int (x^{2} + y^{2}) d m,

I_{z} = I_{x} + I_{y} .

For the uniform ellipse, put

x = a u

y = b v

, where

u^{2} + v^{2} \leq 1

\frac{1}{π} \int_{u^{2} + v^{2} \leq 1} u^{2} d u d v = \frac{1}{4},

and the same for

v^{2}

. Hence

I_{x} = \frac{M b ^{2}}{4}, I_{y} = \frac{M a ^{2}}{4}, I_{z} = \frac{M}{4} (a^{2} + b^{2}) .

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