Newtonian mechanics

Moment of inertia and rotational motion

Moment of inertia I=∫r²dm measures mass distribution about an axis and is the rotational analogue of inertia.

Two equal-mass objects resist angular acceleration differently when one has more mass farther from the axis. Moment of inertia depends on mass and its distance from the axis.

I=\int r_\perp^2dm, \qquad \tau_z=I\alpha, \qquad K_{rot}=\frac12I\omega^2

Definition: Moment of inertia

$r_\perp$ is the perpendicular distance of each mass element from the axis; $I$ is in kg·m². The relation $\tau_z=I\alpha$ applies to a rigid body about a fixed axis. General 3D rotation about changing axes requires the inertia tensor.

Compare a hoop and disk about their symmetry axes; explore torque, angular acceleration, and rotational energy.

Standard results

Body and axis	Moment of inertia
Thin hoop, symmetry axis	$MR^2$
Solid disk, symmetry axis	$MR^2/2$
Thin rod length $L$ , central axis	$ML^2/12$

These formulas assume uniform bodies and the stated axes. The parallel-axis theorem is $I=I_{cm}+Md^2$ for parallel axes offset by $d$ .

Example: A disk under torque

A solid disk has $M=2$ kg, $R=0.30$ m, and net torque $0.18$ N·m. Find its angular acceleration.

Solution

$I=MR²/2=0.09$ kg·m²; $\alpha=\tau/I=2$ rad/s².

Quick check

For a thin hoop, every mass element lies a distance $R$ from the axis, so $I=MR^2$ . In a uniform solid disk, mass lies closer to the axis on average, and integrating over concentric rings gives $I= frac12MR^2$ . Under a constant torque about a fixed axis, angular acceleration is $alpha= au/I$ ; moving mass farther from the axis therefore makes rotation harder to accelerate even at unchanged total mass.

Moment of inertia depends on the selected axis because perpendicular distances $r_perp$ change when the axis changes. The parallel-axis theorem transfers a result from the center-of-mass axis to a parallel one: $I=I_{cm}+Md^2$ . For example, a thin rod of length $L$ about one end has $I=ML^2/3$ , larger than $ML^2/12$ about its center. State both the body and axis before using a formula table.

The unit kg·m² reflects both mass and squared distance from the axis; doubling every distance makes $I$ four times larger. Always state the axis alongside its numerical value.

Which has greater I: a hoop or disk with the same M and R about the symmetry axis?

What rotational equation applies to a rigid body about a fixed axis?

References

John R. Taylor (2005). Classical Mechanics