Moment of inertia of several point masses

For point masses, add the contributions $m_i r_{\perp i}^2$. Here r_i is the shortest perpendicular distance to the axis line, not the distance to an arbitrary point on it. For a z-axis, $r_{\perp i}^2=x_i^2+y_i^2$: moving a particle along z does not change its contribution. A particle on the axis contributes zero in the point model even though its mass is nonzero. For nonnegative masses, contributions are nonnegative; I does not encode rotation direction. The radius of gyration $k=\sqrt{\frac IM}$ represents the total mass imagined at distance k. It conveniently summarises the specified axis, but does not mean actual matter sits there. For a wheel with spokes and rim, mass, density profile and axis all matter; $I=MR^2$ applies only to an ideal thin ring.

A massless rigid frame carries 0.60 kg at 0.20 m, 0.40 kg at 0.35 m, and 1.20 kg on the fixed axis. Its speed is $|\omega|=8.0\,\mathrm{rad\,s^{-1}}$. Square each particle’s own axis distance and add contributions; the on-axis particle has $r_\perp=0$. Use the same I for energy. $I=0.073\,\mathrm{kg\,m^2}$ and E≈2.34 J. The large on-axis mass does not affect these quantities in the point model. Direct check: $v_1=1.60\,\mathrm{m\,s^{-1}}$ and $v_2=2.80\,\mathrm{m\,s^{-1}}$. Their energies 0.768 J and 1.568 J also total 2.336 J; the on-axis particle stays still. First define an axis line and all perpendicular distances; in three-dimensional coordinates, distance to the origin is insufficient. Convert centimetres to metres before squaring: $25\,\mathrm{cm}=0.25\,\mathrm m$, hence $r^2=0.0625\,\mathrm{m^2}$. For one particle, $m=\frac{I}{r_\perp^2}$ for $r_\perp>0$ or $r_\perp=\sqrt{\frac Im}$ for $m>0$. One measured I cannot identify several unknown contributions separately.

The sum defines the instantaneous mass distribution even when components move relative to each other. However, $E=\frac12I\omega^2$ requires shared rigid rotation; additional relative motion contributes its own kinetic energy. If I varies with time, $L_z=I\omega_z$ does not imply simply $\tau_z=I\alpha_z$: differentiation also gives $\dot I\omega_z$. For a freely moving centre of mass, translation must also be separated from rotation. Mass moment of inertia in kg·m² and area second moment in m⁴ are different quantities. A body also has more than one I: axis orientation and location always belong to its specification. In general three-dimensional rotation, an inertia tensor captures couplings that a single number cannot describe.

First define an axis line and all perpendicular distances; in three-dimensional coordinates, distance to the origin is insufficient. Convert centimetres to metres before squaring: $25\,\mathrm{cm}=0.25\,\mathrm m$, hence $r^2=0.0625\,\mathrm{m^2}$. For one particle, $m=\frac{I}{r_\perp^2}$ for $r_\perp>0$ or $r_\perp=\sqrt{\frac Im}$ for $m>0$. One measured I cannot identify several unknown contributions separately. The energy rearrangement requires pure rigid rotation and the same axis; division by ω² is invalid for a stationary body. A small object may be approximated as a particle when its own extent contributes little at the required accuracy. An extended component contributes more accurately $I_{\mathrm{CM,part}}+md^2$, where d is its centre-of-mass distance to the chosen parallel axis. The particle approximation omits intrinsic inertia. For several components, add both contributions.

At equal net torque about a fixed axis, larger I means smaller angular acceleration. At equal angular velocity, it stores more rotational energy. At equal conserved axial angular momentum, it has a smaller angular-speed magnitude. These comparisons hold different quantities fixed: a heavy outer ring therefore need not both rotate faster and store more energy. For the same axis, pure rotation about z also gives $L_z=I_z\omega_z$. With constant I and a fixed inertial axis, $\sum\tau_z=I_z\alpha_z$ follows. These are additional dynamical relations, not requirements for calculating the geometric sum itself. A continuous distribution requires integration over its actual density; a thin line or sheet may instead use linear or surface mass density.