PX155 - E12 - example of moments of intertia

• eg: find $I$ for a thin ring with radius,$R$, and mass ,$M$, about an axis through the centre of the ring, and perpendicular to the ring
  • all mass is at a distance, $R$, from the axis
  • let mass per unit length of circumference $=\sigma$, then, a small section, $d\theta$, will have mass, $dm=R\sigma d\theta$:

- but $2\pi R\sigma =M$,

• eg: find $I$ for a uniform disc with radius, $R$, and mass, $M$, about an axis through its centre, and perpendicular to the disc
  • integrate all rings at radius, $r$, and thickness ,$dr$
  • mass of each ring = $2\pi rdr\sigma$, $\sigma =$ mass per unit area
  • $dm=(2\pi rdr)\sigma$
  • $I=\int r^2.dm=2\pi \sigma {\int }_0^R{r}^3.dr=2\pi \sigma \frac{R^4}{4}$
  • but $\pi R^2\sigma =M$,

• other moments of inertia
  • $I$ for a hollow sphere about an axis through its centre $=\frac{2}{3}M{R}^2$
  • $I$ for a solid sphere about an axis through its centre $=\frac{2}{5}M{R}^2$
  • $I$ for a thin uniform rod, with length, $L$, about a perpendicular axis through its centre $=\frac{1}{12}M{L}^2$
  • $I$ for a thin uniform rod, with length, $L$, about a perpendicular axis through one end $=\frac{1}{3}M{L}^2$
• for exam: $I$ is either given or needs to be derived