AP Physics 1 · Unit 5: Torque and Rotational Dynamics · Lesson 5.4

Deep Dive: Rotational Inertia

🔬 Deep Dive

This is your textbook for this topic. Take your time. Read it more than once.

5.4.A.1Concept

What Is Rotational Inertia?

Rotational inertia (also called moment of inertia, symbol I) is a rigid system's resistance to changes in its rotational motion. It is the rotational analog of mass in linear mechanics.

Just as a larger mass requires a larger force to achieve the same linear acceleration, a larger rotational inertia requires a larger torque to achieve the same angular acceleration. The key difference from mass: rotational inertia depends not just on how much mass a system has, but on where that mass is located relative to the axis of rotation.

🔑Two objects with identical mass can have very different rotational inertias depending on how their mass is distributed. A bowling ball and a hollow sphere of the same mass and size will roll differently because their mass distributions differ — the hollow sphere has all its mass at the surface, giving it larger I.

5.4.A.2Math

I = mr² for a Point Mass

For a single point mass rotating at a perpendicular distance r from an axis:

I = m * r²

The units of rotational inertia are kg·m². The r² factor is what makes mass distribution matter so dramatically. Doubling the distance from the axis quadruples the contribution to rotational inertia — the same quadratic scaling as kinetic energy with velocity.

⚠️r is the perpendicular distance from the axis, not the distance along the object or some other measurement. For a mass on a rod, r is measured straight out from the axis to the mass — perpendicular to the axis of rotation.

ExampleWorked Example — Single Point Mass

A 3 kg ball is attached to a light rod and rotates about an axis 0.8 m from the ball. Calculate its rotational inertia.

5.4.A.3Math

System Rotational Inertia

For a collection of point masses all rotating about the same axis, the total rotational inertia is the scalar sum of individual inertias:

I_tot = Sum(m_i * r_i²) = m_1*r_1² + m_2*r_2² + ...

Each mass contributes independently. The mass with the largest r dominates the total because of r². The AP exam limits this calculation to systems of five or fewer point masses.

Build a system of up to 5 point masses. Each mass contributes I = mr² to the total. Watch how mass farther from the axis dominates the sum.

m12kg @ 0.5m

0.50 kg·m²

m23kg @ 1m

3.00 kg·m²

m31kg @ 1.5m

2.25 kg·m²

Add mass:

m (kg)

r (m)

I_total = Sum(mr²) = 5.75 kg·m²

Add a light mass at large r — watch it dominate the total despite low mass. The r² factor amplifies distance far more than mass alone.

ExampleGuided Example — Three-Mass System

Three point masses are arranged on a light rod: 2 kg at 0.5 m, 4 kg at 1.0 m, and 1 kg at 2.0 m from a common axis. Find I_total.

Step 1 — Calculate each contribution

I_1 = (2)(0.5²) = (2)(0.25) = 0.50 kg·m²
I_2 = (4)(1.0²) = (4)(1.00) = 4.00 kg·m²
I_3 = (1)(2.0²) = (1)(4.00) = 4.00 kg·m²

5.4.A.4Concept

Mass Distribution — Hoop vs. Disk

For extended rigid bodies, the rotational inertia formula must account for the continuous distribution of mass. The results are provided on the AP exam — but understanding why they differ is the real skill.

Thin Hoop

I = MR²

All mass at r = R. Every bit of mass contributes the maximum r². I is the largest possible for given M and R.

Solid Disk

I = ½MR²

Mass spread from r = 0 to r = R. Average r² is smaller. I is exactly half the hoop with same M and R.

Solid Sphere

I = ⅖MR²

Mass distributed in 3D. Even more mass near center than a disk. I is smaller still.

🔑The key ranking: I_hoop > I_disk > I_sphere for the same M and R. More mass farther from the axis → larger I → harder to spin up → rolls slower down a ramp. If you ever see a race between a hoop and a disk rolling down an incline, the disk wins every time.

5.4.A.5Math

Parallel Axis Theorem and Minimum Inertia

A rigid system's rotational inertia is always smallest when the axis passes through the center of mass. This is the minimum inertia rule. For any parallel axis at distance d from the CM axis, the rotational inertia is larger:

I' = I_cm + M * d²

I_cm is the rotational inertia about the center of mass axis, M is the total mass of the system, and d is the perpendicular distance between the two parallel axes. The additional Md² term is always positive — moving the axis away from the CM always increases I.

💡The parallel axis theorem is useful when you know the CM inertia of an object (from the formula sheet) and need to find its inertia about a different parallel axis — such as a rod pivoting at its end rather than its center, or a disk rotating about an edge instead of its center.

Move the axis away from the center of mass and watch rotational inertia increase. At d = 0 the axis passes through the CM — that is always the minimum. Every step away adds Md² to I_cm.

Total mass M2.00 kg

Radius R0.80 m

Axis offset d0.00 m

I_cm (axis through CM)

0.640 kg·m²

minimum possible for this object

Md² (penalty for offset)

0.000 kg·m²

M=2 × d²=0.000

I' = I_cm + Md²

0.640 kg·m²

← at minimum (d = 0)

I_cm

0.64

I' (new)

0.64

Set d = 0 — I' = I_cm, the minimum. Drag d toward R (the edge) and watch I' grow by exactly Md² each time. The hoop has the largest I_cm of the three shapes for the same M and R — so its I' at any offset is always the largest too.

ExampleWorked Example — Rod Rotating About Its End

A uniform rod of mass 2 kg and length 1.2 m has I_cm = (1/12)ML² about its center. Find its rotational inertia about one end.

← Back to Lesson 5.4 Next: Lesson 5.5 →Rotational Equilibrium — the season finale.