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First teaching 2023

First exams 2025

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Moment of Inertia (HL) (HL IB Physics)

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Katie M

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Katie M

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Moment of Inertia

  • In linear motion, the resistance to a change of motion, i.e. linear acceleration, is known as inertia
    • The larger the mass an object has, the greater its inertia
  • In rotational motion, the distribution of mass around an axis must be considered, using moments of inertia
  • The moment of inertia of a rigid, extended body is defined as:

The resistance to a change of rotational motion, depending on the distribution of mass around a chosen axis of rotation

  • Moment of inertia is measured in kg m2
  • The moment of inertia of a body corresponds to how 'easy' or 'hard' it is to rotate, and this is dependent on many factors, including
    • Its shape
    • Its density
    • Its orientation (relative to an axis of rotation)
  • These factors allow an object's distribution of mass to be taken into account 
    • It also means that the moment of inertia of a singular object can change depending on its orientation in relation to the chosen axis of rotation
  • For example, the moment of inertia of a thin rod is different for each of the following orientations:
    • Rotation about its vertical axis
    • Rotation about its centre of mass
    • Rotation about one end

5vz_Kx4c_1-4-5-moment-of-inertia-different-orientations

The moment of inertia of a body can change depending on its orientation relative to the axis of rotation 

  • These are just a few of the possible orientations of the axis of rotation for a thin rod
    • There is an infinite range of possible axes, and therefore moments of inertia
    • This also applies to nearly all rigid, extended objects that could be considered

Examiner Tip

Make sure you are clear on the distinction between linear motion and rotational motion here. The implications of considering the distribution of masses in relation to an axis of rotation, as opposed to considering them as uniform, have important consequences when carrying out calculations

Calculating Moments of Inertia

  • The moment of inertia I of a point mass is equal to

I space equals space m r squared

  • Where:
    • I = moment of inertia (kg m2)
    • m = mass of the object (kg)
    • r = distance from its axis of rotation (m)
  • A point mass is the simplest type of object to consider, the moment of inertia of a non-point mass can be calculated using

I space equals space sum m r squared

  • This means that the sum of the moments of inertia of all the point masses in the system gives the total moment of inertia of the system
  • Some moments of inertia of common shapes are shown below:

1-4-5-moments-of-inertia-for-different-shapes-1

Moments of inertia of common shapes, where R represents radius and L represents length, as shown

Worked example

Two solid spheres form a dumbbell when attached to each end of a thin rod. The dumbbell rotates with the centre of mass of each sphere at a distance of 22 cm from the axis of rotation, as shown in the diagram.

The thin rod has a mass of 20 g. Each sphere has a radius of 4 cm and a mass of 750 g.

1-4-5-moments-of-inertia-dumbbell-arrangement-ib-2025-physics

Moment of inertia of a thin rod about its centre = 1 over 12 m L squared

Moment of inertia of a solid sphere = 2 over 5 m r squared

Determine

(a)
the overall moment of inertia of the dumbbell arrangement
(b)
the ratio of the moment of inertia of the thin rod to the overall moment of inertia of the dumbbell arrangement
 

Answer:

(a)

  • The overall moment of inertia of the dumbbell is the sum of all the moments of inertia in the arrangement

I space equals space sum m r squared space equals space 2 cross times open parentheses 2 over 5 m subscript s p h e r e end subscript r squared close parentheses space plus space 1 over 12 m subscript r o d end subscript L squared

  • Where:
    • Mass of a sphere, m subscript s p h e r e end subscript = 750 g = 0.75 kg
    • Distance from axis to each sphere, r = 22 cm = 0.22 m
    • Mass of the rod, m subscript r o d end subscript = 20 g = 0.02 kg
    • Length of the rod, L = 2 × (22 − 4) = 36 cm = 0.36 m

I space equals space 2 cross times open parentheses 2 over 5 cross times 0.75 cross times 0.22 squared close parentheses space plus space open parentheses 1 over 12 cross times 0.02 cross times 0.36 squared close parentheses

Moment of inertia of the dumbbell:  I = 0.029 kg m2

(b)

  • The moment of inertia of the thin rod is 

I subscript r o d end subscript space space equals space 1 over 12 cross times 0.02 cross times 0.36 squared space equals space 2.16 space cross times space 10 to the power of negative 4 end exponent space kg space straight m squared

  • Therefore, the ratio I subscript r o d end subscript space divided by space I is

I subscript r o d end subscript over I space equals space fraction numerator 2.16 space cross times space 10 to the power of negative 4 end exponent over denominator 0.029 end fraction space equals space 0.0071

  • This means the rod contributes about 0.7% of the overall moment of inertia of the dumbbell

Examiner Tip

You will never be expected to memorise the moments of inertia of different shapes, they will always be given in an exam question where required

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    Katie M

    Author: Katie M

    Expertise: Physics

    Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.