Rotational Inertia

Learning Goals

After this lesson, you should be able to:

• explain why rotational inertia must be stated relative to an axis;
• calculate \(I=mr^2\) for a point mass and sum up to five point masses;
• predict how changing mass or radius changes rotational inertia;
• compare extended rigid systems using provided formulas and mass distribution;
• apply the parallel-axis theorem;
• calculate rotational inertia for a compound system about one common axis;
• interpret useful \(I\)-versus-radius graphs; and
• design an experiment comparing rotational response for different mass distributions.

1. Rotational Inertia Is an Axis-Dependent Property

Mass measures resistance to translational acceleration and does not depend on an axis. Rotational inertia \(I\) measures resistance to angular acceleration and cannot be specified without an axis.

The same rigid body is easier to rotate about some axes than others. A long rod turns more readily about its center than about an end because more of its mass lies farther from the end axis. Rotational inertia exists even while the system is stationary.

3. Point-Mass Definition

For a point mass \(m\) at perpendicular distance \(r\) from an axis,

The distance is the shortest perpendicular distance to the axis, not necessarily the distance to a visible pivot point. A mass directly on the axis has \(r=0\) and contributes zero in the point-mass model.

4. Collection of Point Masses

For rigidly connected point masses, calculate every contribution about the same axis and add:

Rotational inertia is a scalar, so contributions do not cancel even when masses lie on opposite sides of the axis. Direction is irrelevant to \(I\); only mass and perpendicular distance enter.

5. Worked Example: Four-Mass System

Two \(0.50\,\mathrm{kg}\) masses lie \(0.20\,\mathrm m\) from an axis, and two \(1.0\,\mathrm{kg}\) masses lie \(0.40\,\mathrm m\) from it:

Answer: \(I=0.360\,\mathrm{kg\,m^2}\). Although the inner and outer groups have the same total mass, the outer pair contributes eight times as much because it is also twice as massive per object and twice as far from the axis.

6. Functional Dependence

Radius usually has the stronger effect because it is squared. When comparing an entire system, apply the change to only the masses that actually move.

7. Redistributing Mass Without Changing Total Mass

Suppose two equal masses can slide along a light rod. Pulling both inward decreases \(I\); moving both outward increases \(I\). Total mass and center-of-mass position may remain unchanged while rotational inertia changes.

This is the central distinction from translational inertia: knowing only total mass is insufficient. A complete comparison must include the mass distribution and the chosen axis.

8. Extended Rigid Systems

For an extended body, every small mass element contributes according to its distance from the axis. AP Physics 1 provides standard extended-body formulas when numerical use is required; focus on interpreting them.

For equal \(M\) and \(R\), a hoop has greater \(I\) than a disk because more mass is farther out.

9. Parallel-Axis Theorem

If \(I_{\mathrm{cm}}\) is known about an axis through the center of mass, the rotational inertia about a parallel axis a distance \(d\) away is

The axes must be parallel, and \(d\) is the perpendicular distance between them. The added term is nonnegative, so among parallel axes in the same plane, the center-of-mass axis gives the minimum rotational inertia.

10. Worked Example: Rod About One End

A uniform \(3.0\,\mathrm{kg}\), \(1.2\,\mathrm m\) rod has \(I_{\mathrm{cm}}=\frac1{12}ML^2\). Find its rotational inertia about a parallel axis through one end.

Answer: \(I_{\mathrm{end}}=1.44\,\mathrm{kg\,m^2}\), four times the center-axis value.

11. Why the Center-of-Mass Axis Is Minimum

For every axis parallel to the center-of-mass axis,

Equality occurs only at \(d=0\). Shifting the axis moves the system's mass, on average, farther from that axis. This minimum statement applies to the family of parallel axes being compared, not to arbitrary axes in unrelated directions.

12. Compound Systems Need One Common Axis

Add the rotational inertia of each component only after expressing each about the same final axis:

Example: A \(2.0\,\mathrm{kg}\), \(0.30\,\mathrm m\)-radius disk rotates about its center, with a \(0.50\,\mathrm{kg}\) point mass fixed \(0.60\,\mathrm m\) from the same axis:

13. Graphs That Reveal the Model

For one point mass, a graph of \(I\) versus \(r^2\) is linear:

Its slope is the mass and its ideal intercept is zero. By contrast, an \(I\)-versus-\(r\) graph curves upward. For a shifted extended body, a graph of \(I_{\parallel}\) versus \(d^2\) is linear with slope \(M\) and intercept \(I_{\mathrm{cm}}\).

14. Investigation: Change the Mass Distribution

Use a low-friction rotating platform or horizontal rod, two identical movable masses, string and hanging mass or another repeatable torque source, ruler, and video.

1. Place the two masses symmetrically near the axis and measure their radii.
2. Apply the same repeatable torque and record angular position versus time.
3. Move both masses outward without changing total mass or applied torque.
4. Compare the curvature of \(\theta(t)\) or the slope change of \(\omega(t)\) between trials.
5. Calculate the point-mass contribution \(\sum mr^2\) for each arrangement.
6. Test whether the arrangement with larger calculated \(I\) shows the smaller angular response.
7. Discuss axle friction, unequal radii, torque reproducibility, support-rod inertia, frame rate, and release timing.

The experiment tests a comparison; it does not justify ignoring the platform's own rotational inertia.

15. Common Traps and Key Takeaways

• Always state the rotation axis.
• Use perpendicular distance to the axis, not distance to an arbitrary point.
• Radius is squared; doubling it quadruples a point mass's contribution.
• Contributions to \(I\) add and never cancel.
• Rotational inertia exists even at rest.
• Use one common axis before adding compound-body contributions.
• Apply the parallel-axis theorem only to parallel axes.
• Do not memorize every extended-body formula; interpret provided formulas through mass distribution.
• Keep units \(\mathrm{kg\,m^2}\), distinct from torque's \(\mathrm{N\,m}\).