Moment of Inertia of Simple Geometrical Bodies

Deriving the moment of inertia of even a simple uniform body from the defining integral $I = \int r^2\,dm$ (Section 5.10) needs calculus that is beyond WBCHSE's Unit 5 syllabus, which explicitly asks only

for the final VALUES for a short list of simple, common, uniform geometrical bodies about their most

natural axes of symmetry -- to be quoted and used directly, exactly as tabulated below, without needing to

reproduce their derivation.

The single most useful pattern to notice across this whole table is how strongly moment of inertia depends

on exactly how far, on average, a body's mass sits from the stated axis: a thin ring, with all its mass at

the single largest possible distance $R$ from a central axis, has the largest coefficient of $MR^2$ among

comparably sized bodies ($I=MR^2$); a uniform disc, with its mass spread over every radius from $0$ to $R$,

has a noticeably smaller coefficient for the very same axis ($I=\tfrac12 MR^2$, exactly half the ring's

value for equal mass and equal outer radius); and a solid sphere, whose mass is packed still more tightly

around its own centre in three dimensions, has a smaller coefficient again ($I=\tfrac25 MR^2$). The same

comparison holds between a solid cylinder ($\tfrac12 MR^2$, its mass filling the full cross-section) and a

thin hollow cylindrical shell of the same mass and radius ($MR^2$, all of its mass concentrated at the rim)

-- reproducing, for a cylinder, exactly the same solid-versus-hollow contrast already seen between the disc

and the ring.

A rod's moment of inertia about its CENTRE ($\tfrac{1}{12}ML^2$) is noticeably smaller than about one END …