What is Solid; rotation about a fixed axis? — Physics

Definition 19.9 University Physics — Year 1 · Chapter 19 — Systems of Points; Introduction to Rigid Bodies

A rigid body (solid) is a system whose points keep fixed mutual distances. If it turns about an axis $\Delta$ fixed in the frame, every point describes a circle centered on $\Delta$ at the same angular velocity $\omega = \dot\theta$ ; a point at distance $r_i$ from the axis has speed $r_i\omega$ .

Left: the moment of inertia of a disk, summed ring by ring. Right: Huygens’ theorem — about an axis through the rim of a disk, parallel to the axis through its center, J = 1/2 MR2 + MR2. — Left: the moment of inertia of a disk, summed ring by ring. Right: Huygens’ theorem — about an axis through the rim of a disk, parallel to the axis through its center, $J = \tfrac12 MR^2 + MR^2$ .

Three solids in motion: a compound pendulum swinging about a pivot (moment of the weight about the axis), a torsion pendulum (moment of the wire), and a solid rolling without slipping down an incline — the larger its J/MR2, the slower it rolls. — Three solids in motion: a compound pendulum swinging about a pivot (moment of the weight about the axis), a torsion pendulum (moment of the wire), and a solid rolling without slipping down an incline — the larger its $J/MR^2$ , the slower it rolls.

Examples

Example 19.12 (Compound pendulum; torsion pendulum)

A solid of mass $M$ pivoting about a horizontal axis at distance $d$ from $G$ : the weight’s moment is $-Mgd\sin\theta$ , so $J_\Delta\ddot\theta = -Mgd\sin\theta$ and small swings have the period

T = 2\pi\sqrt{\frac{J_\Delta}{Mgd}} ,

that of a simple pendulum of length $J_\Delta/Md$ — for a uniform rod pivoted at one end, $2L/3$ . A disk hung from a wire of torsion constant $C$ (restoring moment $-C\theta$ ): $J\ddot\theta = -C\theta$ , $T = 2\pi\sqrt{J/C}$ — the torsion balance that measured $G$ and Coulomb’s law.

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