🔄Rotational Inertia
The Rotational Analog of Mass
Rotational inertia (moment of inertia, I) is the resistance to changes in rotational motion. Just as mass resists changes in linear motion, rotational inertia resists changes in angular motion.
I = Σ mr²
For a point mass: I = mr² The farther the mass from the rotation axis, the larger I.
Common formulas: - Solid disk: I = ½MR² - Ring (hoop): I = MR² - Solid sphere: I = (2/5)MR² - Thin rod (center): I = (1/12)ML² - Thin rod (end): I = (1/3)ML²
Notice: a hoop has larger I than a disk of the same mass and radius — because all the mass is at the outer edge.
Think About It
Two cylinders of the same mass and radius roll down a ramp: one is solid, one is hollow (like a pipe). Which reaches the bottom first? Why?
✏️ Worked Example
Problem: A torque of 20 N·m acts on a solid disk (M = 5 kg, R = 0.4 m). Find the angular acceleration.
📐 Key Equations
Rotational Inertia
I = Σ mr²I_disk = (1)/(2)MR², I_hoop = MR²I_sphere = (2)/(5)MR²τ = Iα⚠️ Common Mistakes
Misconception: Rotational inertia depends only on mass, not on how the mass is distributed.
✓ Correct thinking: I = Σmr² — mass farther from the rotation axis contributes more to I than mass near the axis.
Why: A hoop (I = MR²) is harder to spin than a disk (I = ½MR²) of identical mass and radius because all the hoop's mass sits at the maximum distance.
Misconception: The same object always has the same rotational inertia.
✓ Correct thinking: Rotational inertia depends on which axis you choose.
Why: A thin rod has I = (1/12)ML² about its center but I = (1/3)ML² about its end — different axis, very different I.
Misconception: A heavier object always has greater rotational inertia than a lighter one.
✓ Correct thinking: Both mass AND distribution matter. A less massive ring can have greater I than a heavier solid disk if the ring's radius is large enough.
Why: I scales with mr², so radius has an outsized effect (squared term). A small mass far out beats a large mass close in.
📝 Practice Problems
Try these problems. Check your answer when ready.
Calculate the rotational inertia of a solid disk with M = 8 kg and R = 0.5 m.
I_disk = (1)/(2)MR²A hoop and a solid disk have the same mass (3 kg) and radius (0.4 m). Which has the greater rotational inertia, and by how much?
A net torque of 12 N·m acts on a solid cylinder (M = 6 kg, R = 0.2 m). Find the angular acceleration.
α = τ / ITwo point masses m = 2 kg are attached to the ends of a massless rod 1.2 m long. Find I about (a) the center and (b) one end of the rod.
A solid sphere (M = 5 kg, R = 0.3 m) rolls without slipping. Find its rotational inertia.
I_sphere = (2)/(5)MR²A solid disk (I = ½MR²) starts from rest and reaches ω = 20 rad/s in 5 s under a constant torque of 8 N·m. Find M if R = 0.4 m.
Compare the times for a solid disk and a hollow ring (same M and R) to reach the same angular speed from rest under the same applied torque.
Finished reading through this lesson?