The moment of inertia or mass moment of inertia is a scalar quantity that measures a rotating body’s resistance to rotation. The higher the moment of inertia, the more resistant a body is to angular rotation. A body is usually made from several small particles forming the entire mass. The mass moment of inertia depends on the distribution of each individual mass concerning the perpendicular distance to the axis of rotation. However, in physics, we usually assume that the mass of an object is concentrated at a single point called the centre of mass.
Moment of inertia equation
Mathematically, the moment of inertia can be expressed in terms of its individual masses as the sum of the product of each individual mass and the squared perpendicular distance to the axis of rotation. You can see this in the equation below. I is the moment of inertia measured in kilogram square metres (kg·m2), m is the mass measured in kilograms (kg), and r is the perpendicular distance to the axis of rotation measured in metres (m).
\[I = \sum_i^n m \cdot r^2_i\]
We can also use the equation below for an object whose mass is assumed to be concentrated to a single point. The image shows the distance of the axis of rotation r.
Fig. 1 - Diagram showing the distance of the axis of rotation r
\[I = m \cdot r^2\]
Where did the moment of inertia come from?
Newton’s law states that the linear acceleration of an object is linearly proportional to the net force acting upon it when mass is constant. We can state this with the equation below, where Ft is the net force, m is the object’s mass, and at is the translational acceleration.
\[F_t = m \cdot a_t\]
Similarly, we use torque for rotational motion, which is equal to the product of the rotational force and the perpendicular distance to the axis of rotation. However, the translational acceleration for rotational motion is equal to the product of angular acceleration α and radius r.
\[\alpha_t = r \cdot \alpha \frac{T}{r} = m \cdot r \cdot \alpha \Rightarrow T = m \cdot r^2 \cdot \alpha\]
The moment of inertia is the reciprocal of the mass in Newton’s second law for linear acceleration, but it is applied to angular acceleration. Newton’s second law describes the torque acting on a body, which is linearly proportional to the mass moment of inertia of a body and its angular acceleration. As seen in the derivation above, the torque T is equal to the product of the moment of inertia I and angular acceleration \(\alpha\).
\[T = I \cdot \alpha \]
Moments of inertia for different shapes
The moment of inertia is different for and specific to each object’s shape and axis. Due to the variation in geometric shapes, a moment of inertia is given for various commonly used shapes, which you can see in the image below.
Fig. 2 - Moment of inertia for different shapes
We can calculate the moment of inertia for any shape by integration (about the x-axis) of the product of the equation, which describes the width or thickness d, the rate of change of y, and A multiplied by the squared distance to the axis.
\[I = \int dA \cdot y^2\]
The greater the thickness, the greater the moment of inertia.
Examples of calculating the moment of inertia
A thin disk with a 0.3 m diameter and a total moment of inertia of 0.45 kg·m2 is rotating about its centre of mass. There are three rocks with masses of 0.2 kg on the outer part of the disk. Find the total moment of inertia of the system.
Solution
The radius of the disk is 0.15 m. We can calculate the moment of inertia of each rock as
\[I_{rock} = m \cdot r^2 = 0.2 kg \cdot 0.15 m^2 = 4.5 \cdot 10^{-3} kg \cdot m^2\]
Hence, the total moment of inertia will be
\[I_{rocks} + I_{disk} = (3 \cdot I_{rock})+I_{disk} = (3 \cdot 4.5 \cdot 10^{-3} kg \cdot m^2) + 0.45 kg \cdot m^2 = 0.4635 kg \cdot m^2\]
An athlete is sitting in a rotating chair holding a training weight of 10kg in each hand. When will the athlete be more likely to rotate: when he extends his arms far from his body or when he retracts his arms close to his body?
Solution
When the athlete extends his arms, the moment of inertia increases as the distance between the weight and his axis of rotation increases. When the athlete retracts his arms, the distance between the weights and the axis of rotation decreases, and so does the moment of inertia.
Therefore, the athlete is more likely to rotate when he retracts his hands as the moment of inertia will be smaller and the body will have less resistance to rotating.
A very thin disk with a diameter of 5cm is rotating about its centre of mass, and another thicker disc with a diameter of 2 cm is rotating about its centre of mass. Which of the two disks has a larger moment of inertia?
Solution
The disc with the larger diameter will have a larger moment of inertia. As the formula suggests, the moment of inertia is proportional to the squared distance to the axis of rotation, hence the greater the radius, the larger the moment of inertia.
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