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5.4Torque and Rotational Dynamics

Rotational Inertia — AP PHYSICS 1 Revision Notes & Topical Questions

5 min read 9 parts Lesson 14 of 18 1 quiz

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Lesson summary

5.4 Rotational Inertia — AP PHYSICS 1 notes: Describe the rotational inertia of a rigid system about a given axis, including axes not through the centre of mass.Rotational inertia = the resistance of a rigid system to changes in rotational motion, depending on its mass and how th…

  • Dependence
  • Mass distribution
  • Point mass
  • Same mass and radius
  • Hollow versus solid cylinder
  • I = Icm + Md2

Section 1 of 9Rotational inertia

Rotational inertia
Rotational inertia the resistance of a rigid system to changes in rotational motion, depending on its mass and how that mass is distributed about the chosen axis.
Role
  • Role it plays the part of mass in rotation: τnet = Iα compares with Fnet = ma.
Unit
  • Unit kg·m2.
Dependence
  • Dependence shape, density and orientation relative to the axis;
    • the same object has different I about different axes.
Mass distribution
  • Mass distribution mass farther from the axis contributes much more, in proportion to r2.

Section 2 of 9Point masses and systems

Point masses and systems
Point mass
  • I = mr2, r the perpendicular distance from the axis.
System
  • Itotal = Σmiri2; parts rotating about the same axis simply add.
Example
  • Example two masses m at r and two masses 2m at 2r give 2mr2 + 16mr2 = 18mr2, with the upper pair contributing 11%.

Section 3 of 9Standard rigid bodies

ObjectAxisI
Thin hoop (ring)CentreMR2 (largest)
Solid disk or cylinderCentre½MR2
Solid sphereCentre(2/5)MR2 (smallest)
Thin rodCentre(1/12)ML2
Thin rodEnd(1/3)ML2
1Same mass and radius
  • Same mass and radius hoop > disk > solid sphere because more hoop mass is at the rim.
2Hollow versus solid cylinder
  • Hollow versus solid cylinder the hollow one has more I as its mass is farther from the axis.

Section 4 of 9Parallel axis theorem

I
Icm + Md2
  • d is the distance between the centre-of-mass axis and a parallel axis.
Minimum
  • I is smallest about an axis through the centre of mass.
Equal distances
  • Equal distances all parallel axes at the same distance give the same I.

Section 5 of 9Changing the shape

Changing the shape
Skater pulling in arms
  • I changes although mass is unchanged.
Tightrope walker's pole
  • Tightrope walker's pole increases I.
Hoop and sphere rolling down an incline
  • Hoop and sphere rolling down an incline the hoop rolls down more slowly.

Section 6 of 9Finding the rotational inertia of a system

1
Identify the axis of rotation.
2
Split the system into point masses and standard shapes.
3
For each point mass, calculate mr2 with the perpendicular distance to the axis.
4
For each extended shape, use its standard I, adding Md2 if the axis is not through its centre of mass.
5
Add all contributions to find the total I.

Section 7 of 9More from the specification

1Moment of inertia of rigid bodies
  • Moment of inertia of rigid bodies mass distribution relative to rotational axes determines angular acceleration resistance in rotating solid rigid bodies.
2Rigid body
  • Body whose shape does not change when rotating.
3Length
  • Resistance is directly proportional to the length of a wire.
4Linear
  • A graph that is a straight line through the origin, so resistance is constant.

Section 8 of 9Key terms

TermMeaning
Rotational inertiaResistance to angular acceleration, I = Σmr2.
Mass distributionMass farther from the axis increases I.
Point massMass with I = mr2.
Parallel-axis theoremI = Icm + Md2.
kg·m2Unit of rotational inertia.

Section 9 of 9Exam technique

1
Using the object's total mass at the centre instead of the distance of each part from the axis.
2
Forgetting that I depends on the chosen axis.
3
Using r instead of r2.
4
Using the same formula for hoops, disks and spheres.

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