Rigid Body Dynamics: Moment of Inertia Theorems

Rigid Body Definition

A rigid body is a special case of a many-particle system. It is defined by the condition that the distance between the particles of the body remains constant ($R = \text{constant}$), meaning the body is absolutely non-deformable.

Steiner's Theorem (Parallel Axis Theorem)

We can calculate the moment of inertia of a rigid body ($I_O$) about a rotation axis passing through a point $O$, provided we know the moment of inertia about an axis parallel to the first and passing through the center of mass ($I_{cm}$).

This relationship is:

I_O = I_cm + M d²

Where:

• $I_O$ is the moment of inertia of the body about the new axis (passing through $O$).
• $I_{cm}$ is the moment of inertia about the axis through the center of mass ($C$).
• $M$ is the total mass of the body.
• $d$ is the perpendicular distance between these two parallel axes.

Derivation of Steiner's Theorem

If the moment of inertia of the solid about an axis through $O$ is $I_O = \Sigma m r²$, and the moment of inertia about an axis passing through $C$ is $I_C = \Sigma m R²$.

Relating $r$ and $R$ by the expression derived from geometry: $r² = R² + d² + 2 d x_c$.

Substituting this into the definition of $I_O$ yields:

I_O = I_C + M d² + 2 d \Sigma m x_c

The middle term on the right side ($2 d \Sigma m x_c$) is zero because $x_c$ represents the position relative to the center of mass, and the first moment of mass about the center of mass is zero.

Thus, $I_O = I_C + M d²$.

Methods for Determining Moment of Inertia

The moment of inertia of a given body can be determined by knowing:

1. Symmetry: The symmetry of the body can sometimes simplify the calculation significantly.
2. Additivity: Since the moment of inertia is additive, the calculation for a composite body can be taken as the sum of the moments of inertia of its constituent parts.
3. Steiner's Theorem: The moment of inertia of a body about a specific axis can often be found using the moment about another parallel axis via Steiner's theorem.

Perpendicular Axes Theorem

The moment of inertia of a thin plate (or planar body) about an axis ($z$) perpendicular to the plate is equal to the sum of the moments of inertia about two axes ($x$ and $y$) which are contained within the plane of the plate, provided that all three axes are mutually perpendicular and intersect at a common origin.

The relationship is:

I_z = I_x + I_y

The Perpendicular Axes Theorem applies only to plane figures and allows us to relate the moment perpendicular to the plane with the moments of the other two axes lying in the plane.

Oscillator Equations

Driven Oscillator Equations

Driving Term: $\Gamma$ or $\sin(\omega t)$. $\Gamma_0 = F_0 / m$.

Resonance Factor: $\omega R = \sqrt{\omega^2 \text{ or } \omega_0^2 - 2\gamma}$ (Note: This formula structure is highly unconventional and likely incomplete or corrupted in the source text.)

Damped Oscillator Equations

The differential equation for the damped oscillator is:

d²x / dt² + 2$\gamma$ dx / dt + $\omega_0$² x = 0

Related parameters:

$\gamma = \sqrt{k/m}$

Quality Factor: $Q = \omega / 2\gamma$