IB Physics HLRigid Body MechanicsHL onlyPaper 1 & 2~11 min read
Moment of Inertia
In straight-line motion, an object’s mass tells you how hard it is to get moving. In rotation, there’s an equivalent quantity — but it depends on more than just how much mass there is. It also depends on where that mass sits relative to the axis of spin. Mass spread out far from the axis is much harder to rotate than the same mass tucked in close. This “rotational mass” is the moment of inertia, and it’s the key that unlocks Newton’s second law for rotation.
📘 What you need to know
Moment of inertia is the resistance to a change in rotational motion
It depends on the mass and how that mass is distributed around the axis of rotation
It’s measured in kg m2
For a single point mass: I = mr2
For a system of masses: Itot = Σmr2 (add up each part)
The same object can have different moments of inertia about different axes
Moments of inertia for standard shapes are always given in exam questions — you don’t memorise them
What is moment of inertia?
The moment of inertia of a rigid, extended body is defined as the resistance to a change in rotational motion, depending on the distribution of mass around a chosen axis of rotation. It plays the same role in rotation that mass plays in linear motion — it tells you how hard it is to change the spin.
Crucially, it depends not just on how much mass there is, but on how far that mass is from the axis. Mass far from the axis contributes far more, because the distance is squared. That’s why the same thin rod is much easier to spin about its length than about its centre:
The same thin rod has a different moment of inertia about each axis. Spinning about its own length is easiest; about one end is hardest, because the mass is furthest from the axis.
Think of a figure skater or a diver. Same body, same mass — but tuck the arms and legs in and they spin much more easily than when spread out. That’s moment of inertia in action: pulling mass closer to the axis lowers I. The mass hasn’t changed, only its distribution, and because distance is squared, even a small change makes a big difference.
Calculating moments of inertia
The building block is the moment of inertia of a single point mass — a mass m at a distance r from the axis:
Moment of inertia of a point massI = mr2
where I is the moment of inertia in kg m2, m is the mass in kg, and r is the distance from the axis in metres. For a system made of several masses, the total moment of inertia is just the sum of each part’s contribution:
Total moment of inertia of a systemItot = Σmr2
WE 1
A 0.50 kg point mass is fixed to the end of a light rod, 0.30 m from the axis of rotation. Calculate its moment of inertia.
Step 1 — use I = mr²
I = m × r²
Step 2 — substituteI = 0.50 × 0.30² = 0.50 × 0.09I = 0.045 kg m²Because r is squared, doubling the distance would quadruple the moment of inertia.
WE 2
Two point masses are fixed to a light rotating frame: a 2.0 kg mass 0.40 m from the axis and a 3.0 kg mass 0.25 m from the axis. Calculate the total moment of inertia.
Step 1 — add each mass’s contribution: Itot = Σmr²
Itot = m₁r₁² + m₂r₂²
Step 2 — substituteItot = (2.0 × 0.40²) + (3.0 × 0.25²)Itot = 0.32 + 0.1875Itot = 0.51 kg m² (2 s.f.)Add the moments of inertia, not the masses — each mass’s distance matters separately.
Moments of inertia of common shapes
Real objects aren’t point masses, so their moments of inertia come from integrating over their shape. You never need to derive or memorise these — they’re always provided in exam questions. But it helps to recognise the pattern: each is a numerical fraction × m × (a radius or length)2.
A few common moments of inertia. Each is a fraction × mass × (radius or length)2. These are always given in exam questions — you just substitute.
WE 3
Two solid spheres form a dumbbell on a thin rod. The rod has mass 150 g, and each sphere (radius 4 cm) has a moment of inertia of 0.04 kg m2 about the axis. The sphere centres are 22 cm from the axis. Using Irod = (1/12)mL2, find the total moment of inertia.
Step 1 — find the rod length (centre to centre minus sphere radii)L = 2 × (0.22 − 0.04) = 0.36 mStep 2 — moment of inertia of the rodIrod = (1/12) × 0.15 × 0.36² = 1.62 × 10⁻³ kg m²Step 3 — total = two spheres + rodItot = 2(0.04) + 1.62 × 10⁻³Itot = 0.082 kg m²The rod contributes only about 2% — almost all the inertia is in the spheres, out at the ends.
Point mass I = mr²
add up →
System Σmr²
or use →
Shape formula given in Q
🛠️ Finding a moment of inertia
Identify the axis of rotation — the same object has different I about different axes.
Point masses? Use I = mr2 for each, then add: Itot = Σmr2.
Standard shape? Use the formula given in the question and substitute mass and radius/length.
Combined objects? Add the moments of inertia of each part (never the masses).
Keep distances in metres so I comes out in kg m2.
💡 Top tips
Never memorise shape formulas. They’re always provided — your job is to substitute correctly.
Add moments, not masses. For a system, sum each part’s mr2 separately.
Distance is squared. Mass far from the axis dominates — a small distance change has a big effect.
Axis matters. Always check which axis the formula (and the question) refers to.
Quick recap: Moment of inertia is rotational “mass” — the resistance to changing rotation, in kg m2. It depends on both the mass and its distribution about the axis. For a point mass I = mr2; for a system Itot = Σmr2; and standard-shape formulas are always given in exams.
⚠ Common mistakes
Adding the masses of a system instead of adding each part’s moment of inertia
Forgetting that r is squared in I = mr2
Using the wrong axis — the same object has different I depending on the axis
Trying to memorise shape formulas instead of using the one given
Leaving distances in cm instead of converting to metres
You’ve now got the rotational version of mass. That’s the last piece needed to write down Newton’s second law for rotation: torque equals moment of inertia times angular acceleration, τ = Iα — the rotational mirror of F = ma. That’s exactly what we tackle next.
Want this to actually click before the exam?
Book a free meeting and let’s work through the tricky bits together.
