Foundations — Physical pendulum — compound pendulum

This page assumes nothing. If the parent note wrote a symbol without telling you where it came from, we build it here from a picture first. Read top to bottom — each block uses only what the block above it already defined.

1. What "rigid body" and "pivot" mean

Before any maths, picture the object itself.

A rigid body is a solid object whose parts never move relative to each other — a ruler, a door, a wheel. It keeps its shape while it swings. This is the opposite of a floppy string.

The ==pivot $O$== is the fixed line (a nail, a rod, an axle) the body turns around. In the picture it is the black dot at the top; the body can only rotate about it — it cannot slide away. Every point of the body traces a circle centred on $O$.

2. Angle $\theta$ — measuring "how far tilted"

When the body hangs still, its balance-point sits straight below the pivot. This resting position is called equilibrium.

Push it aside and it makes an angle with that vertical line. We call this angle ==$\theta$== (Greek letter "theta"), the angular displacement.

• $\theta =0$ means hanging straight down (equilibrium).
• $\theta$ positive means tilted one way, negative the other way — the sign just records the direction of the tilt.

3. Center of mass and the distance $d$

Every object has one special balance-point: the center of mass (CM). If you supported the object exactly there, it would balance perfectly, and gravity behaves as if all the weight acts at this one point.

The symbol ==$d$== is the straight-line distance from the pivot $O$ to the CM.

For a uniform ruler the CM is at its middle, so if you pivot at one end, $d$ = half the length. For the disk in the parent note, the CM is the disk's center, so pivoting at the rim gives $d=R$.

4. Mass $m$ and weight $mg$

• ==$m$== is the mass — how much stuff the object is made of, in kilograms.
• ==$g$== is the acceleration due to gravity, about $9.8{\text{m/s}}^2$.
• The product ==$mg$== is the weight — the downward force gravity exerts. It always points straight down, and (from §3) it acts at the CM.

5. Torque $\tau$ — the "twisting strength" of a force

A force that pushes straight toward the pivot cannot spin anything. Only the sideways part twists. ==Torque $\tau$== (Greek "tau") measures how strongly a force twists a body about an axis.

$\tau =(\text{force})\times (\text{perpendicular distance from the axis to the force’s line})$

For our pendulum the force is $mg$ (down) and the CM sits a distance $d$ from the pivot. When tilted by $\theta$, the perpendicular lever arm of that downward force is $d\sin \theta$ (the horizontal offset of the CM from the pivot). So

$\tau =-mgd\sin \theta .$

6. Moment of inertia $I$ — "rotational heaviness"

Here is the star of the topic. ==Moment of inertia $I$== measures how hard it is to change a body's spin — the rotational version of mass. Its size depends not just on how much mass there is, but on how far that mass sits from the axis:

$I=\sum (\text{each bit of mass})\times (\text{its distance from the axis}{)}^2.$

Distant mass counts far more (the distance is squared), so a body with mass spread wide is much harder to spin.

The link Moment of inertia has the full derivation of the standard values ($\frac{1}{3}m{\ell }^2$ for a rod-about-end, $\frac{1}{2}m{R}^2$ for a disk, ...).

7. Radius of gyration $k$ — packaging $I$ into a length

Sometimes we want $I$ expressed as a distance. The ==radius of gyration $k$== is defined so that

$I_{cm}=m{k}^2⟺k=\sqrt{\frac{I_{cm}}{m}}.$

Think of it as: "if all the mass were squeezed onto a thin ring of radius $k$, it would have the same $I_{cm}$." It repackages the mass-spread as a single length. This lets the parent note rewrite $L_{\text{eq}}=k^2/d+d$ neatly and find the minimum-period pivot at $d=k$. Full detail: Radius of gyration.

8. Angular acceleration and the rotational Newton's law

• $\frac{d\theta }{dt}$ = how fast the angle changes = angular velocity.
• $\ddot{\theta }=\frac{d^2\theta }{d{t}^2}$ = how fast the angular velocity changes = angular acceleration $\alpha$ (the two dots just mean "rate of change, twice").

The rotational Newton's second law ties torque to spin-up:

$\tau =I\ddot{\theta }.$

Compare $F=ma$: torque $\tau$ plays the role of force, moment of inertia $I$ plays the role of mass, and angular acceleration $\ddot{\theta }$ plays the role of ordinary acceleration. This is the equation the parent note feeds the torque into.

9. The SHM signature — why $\ddot{\theta }=-{\omega }^2\theta$ means "swing"

Combine §5 and §8, then use the small-angle shortcut $\sin \theta \approx \theta$ (valid for small tilts, in radians):

$I\ddot{\theta }=-mgd\sin \theta \approx -mgd\theta \Rightarrow \ddot{\theta }=-\frac{mgd}{I}\theta .$

Any equation of the shape $\ddot{\theta }=-{\omega }^2\theta$ is Simple Harmonic Motion: the acceleration always points back toward zero, in proportion to how far you are from zero. That is exactly what makes something oscillate with a steady period. Matching the two forms gives

${\omega }^2=\frac{mgd}{I},T=\frac{2\pi }{\omega }=2\pi \sqrt{\frac{I}{mgd}}.$

• ==$\omega$== ("omega") = angular frequency, radians of the cycle per second.
• ==$T$== = period, seconds for one complete back-and-forth. They are linked by $T=2\pi /\omega$ because one full cycle is $2\pi$ radians of phase.

10. How every foundation feeds the topic

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