Unit 5 · Topic 5.4 Beta

Rotational Inertia

Rotational inertia I measures how hard an object is to set turning about an axis.

Practice 2: Mathematical RoutinesPractice 3: Scientific Questioning and Argumentation

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Question set for this topic

Part 1 · Hook

Why this matters

A figure skater, a tightrope walker’s long pole and a baseball bat held by the wrong end all show the same thing. How hard something is to set turning depends on where its mass sits, not just how much there is. Move the same mass farther from the axis and it fights a change in rotation much harder.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. If r doubles, r² becomes

  1. 4 times as large
  2. 2 times as large
  3. half as large
  4. the same
Show the answer

(2r)² = 4r².

  • Correct: 4 times as large:
  • 2 times as large:
  • half as large:
  • the same:

2. The center of mass of two equal masses is

  1. halfway between them
  2. at the heavier one
  3. at either end
  4. outside the line joining them
Show the answer

Equal masses balance at the midpoint.

  • Correct: halfway between them:
  • at the heavier one:
  • at either end:
  • outside the line joining them:

Part 4 · See it

See it first

Two light rods each carry two equal masses m and turn about a vertical axis through their centers. In the top rod the masses sit a distance r from the axis, giving I = 2mr². In the bottom rod they sit at 2r, giving I = 8mr², four times as much with the same total mass. A panel gives I = Σmr², notes that doubling the distance quadruples I, and lists rotational inertias the exam gives for shapes: hoop MR², solid disk ½MR², solid sphere two fifths MR², rod about its center one twelfth ML², rod about one end one third ML². Unit kg·m².
The same two masses at r and at 2r: twice the distance gives four times the rotational inertia. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. Every bit of mass in a turning object moves in a circle about the axisit resists changes in that motion in proportion to its mass
  2. Mass farther from the axis moves faster for a given ωit counts more, as the square of its distance
  3. Adding mr² for every bit of massgives the rotational inertia I = Σmr², in kg·m²
  4. The same mass is moved farther from the axisI increases (with r²)
  5. The axis is moved a distance d from the center of massI = I_cm + Md², always larger than I_cm

Part 6 · Key ideas

Key ideas

  • Rotational inertia I = Σmr² is the rotational partner of mass: it measures resistance to a change in rotation.
  • r is each mass’s distance from the axis, and it is squared: doubling r gives 4 times the I.
  • Same mass, different distribution, different I. A hoop (MR²) beats a disk (½MR²) of the same M and R.
  • Inertias of disks, rods, hoops and spheres are given on the exam; you do not memorize them.
  • Parallel-axis theorem: I = Icm + Md². I is least about the center of mass.

Part 7 · Misconception

A common mistake

The wrong idea: Two objects with the same mass have the same rotational inertia.

What actually happens: Rotational inertia depends on where the mass is. A hoop and a disk of equal mass and radius differ by a factor of 2, and the same rod is 4 times harder to turn about its end than about its middle.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Diagram

A rod with three balls

0.20 kg0.40 kg0.20 kgaxis B (end)axis A (center)0.40 m0.40 mLight rod 0.80 m long; treat each ball as a point object. Each axis is perpendicular to the rod.

1. What is the rotational inertia of the rod and balls about axis A? Give the value and its unit.

Type a number and its unit.

Show the answer

The middle ball is on axis A (r = 0). I = 2(0.20 kg)(0.40 m)² + 0 = 0.064 kg·m².

  • Answer: 0.064 kg·m^2

2. What is the rotational inertia about axis B?

Type a number in kg·m^2.

Show the answer

About B: the near ball has r = 0, the middle ball r = 0.40 m, the far ball r = 0.80 m. I = 0 + (0.40 kg)(0.40 m)² + (0.20 kg)(0.80 m)² = 0.064 + 0.128 = 0.192 kg·m², about 0.19 kg·m².

  • Answer: 0.19 kg·m^2

3. Why is the rotational inertia about axis B larger than about axis A, although the mass is the same?

  1. The mass is, on average, farther from axis B
  2. Axis B is at the end, where the rod is heavier
  3. Axis B passes through less of the mass
  4. Rotational inertia is larger about any axis at an end
Show the answer

I = Σmr². About B the 0.20 kg ball is 0.80 m away, and r² makes that distant mass count four times as much as at 0.40 m. Same mass, spread farther from the axis: larger I.

  • Correct: The mass is, on average, farther from axis B: Right: I depends on how far the mass is from the axis, as r².
  • Axis B is at the end, where the rod is heavier: The rod is light; its ends are not heavier. The distances to the axis change.
  • Axis B passes through less of the mass: Axis B passes through a 0.20 kg ball; axis A through the 0.40 kg ball. What matters is how far the rest is.
  • Rotational inertia is larger about any axis at an end: An end axis gives a larger I here because of the distances, not because it is an end.

4. The two end balls are slid in to 0.20 m from the middle. The rotational inertia about axis A becomes

  1. one quarter as large
  2. half as large
  3. the same
  4. twice as large
Show the answer

Only the end balls contribute about A, each as mr². Halving r multiplies r² by ¼, so I goes from 0.064 to 0.016 kg·m².

  • Correct: one quarter as large: Right: r halves, so each r² term is ¼ as large.
  • half as large: I depends on r², not r.
  • the same: The mass is the same, but it is closer to the axis.
  • twice as large: Moving mass toward the axis decreases I.

5. The rotational inertia of an object about an axis depends on

  1. its mass and how that mass is spread around the axis
  2. its mass alone
  3. how fast it is turning
  4. the forces acting on it
Show the answer

I = Σmr²: each bit of mass counts with the square of its distance from the axis, so both the mass and its distribution matter.

  • Correct: its mass and how that mass is spread around the axis: Right: I = Σmr².
  • its mass alone: Two objects of the same mass can have very different I.
  • how fast it is turning: I is a property of the mass and the axis, not of the motion.
  • the forces acting on it: Forces give torques; they do not change I.

6. A thin hoop and a solid disk have the same mass and radius. Which is harder to start spinning about its center, and why?

  1. The hoop, because all its mass is at the rim
  2. The disk, because it is solid and has more material
  3. Neither, because they have the same mass
  4. The disk, because its mass is closer to the axis
Show the answer

Rotational inertia measures resistance to changes in turning. The hoop’s mass sits at the rim (I = MR²); much of the disk’s mass is near the center (I = ½MR²), so the hoop is harder to spin up.

  • Correct: The hoop, because all its mass is at the rim: Right: the hoop’s mass is at r = R, so I = MR² > ½MR².
  • The disk, because it is solid and has more material: Same mass: the disk’s mass is spread inward, so its I is smaller.
  • Neither, because they have the same mass: Same mass, different distribution, different rotational inertia.
  • The disk, because its mass is closer to the axis: Mass close to the axis makes an object easier to spin, not harder.

7. For a given object, about which of these parallel axes is the rotational inertia smallest?

  1. The axis through its center of mass
  2. An axis through its edge
  3. An axis halfway between the center and the edge
  4. It is the same about each parallel axis
Show the answer

I = I_cm + Md², and Md² is zero only when the axis passes through the center of mass, so that axis gives the smallest I.

  • Correct: The axis through its center of mass: Right: I = I_cm + Md² is least when d = 0.
  • An axis through its edge: Moving the axis away from the center of mass adds Md².
  • An axis halfway between the center and the edge: This still adds Md² with d > 0.
  • It is the same about each parallel axis: The parallel-axis theorem shows I grows with d².

Part 9 · Summary

Summary

Rotational inertia I measures how hard an object is to set turning about an axis. For point masses I = Σmr², so it depends on both the mass and how far the mass is from the axis, with distance counted squared. The inertias of standard shapes are given. About an axis a distance d from the center of mass, I = I_cm + Md².

Part 10 · Up next

What comes next

Part 11 · Connections

Connections