1. What is Moment of Inertia?
Moment of inertia (MI) tells us how difficult it is to change the rotational motion of a body. Just as mass resists changes in linear motion, moment of inertia resists changes in rotational motion. A body with a larger moment of inertia needs a greater torque to start rotating or to stop rotating.
You already experience this concept in daily life: spinning a long rod is harder than spinning a small stone, even if both have the same mass. This happens because their mass is distributed differently.
1.1. Everyday Intuition
- A door opens easily when you push far from the hinges, but feels heavy near the hinge. This is because the distribution of mass affects how easily it rotates.
- A figure skater spins faster by pulling their arms in — reducing moment of inertia.
- A bicycle wheel rotates more easily than a solid disc of the same size because its mass is mostly near the rim.
1.1.1. Key Insight
Moment of inertia is not just about how much mass an object has — it is about how that mass is spread out relative to the axis of rotation.
2. Definition and Formula
Moment of inertia depends on both mass and the distance of mass from the axis of rotation. For a system of particles, the MI is given by:
\( I = \sum_i m_i r_i^2 \)
Here:
- \( m_i \) is the mass of each particle
- \( r_i \) is the distance of the particle from the rotation axis
2.1. Continuous Mass Distribution
For solids like rods, discs, or spheres, we treat mass as distributed continuously. The formula then becomes an integral:
\( I = \int r^2 \, dm \)
This allows us to calculate MI for different shapes, which we will explore in related topics.
2.2. Units and Dimensions
The SI unit of moment of inertia is kg·m². Its dimensional formula is \( [M L^2] \).
3. Dependence on Mass Distribution
The moment of inertia increases when more mass is located farther from the rotation axis. Even objects with the same mass can have different MI values depending on how their mass is arranged.
3.1. Examples to Understand Distribution
- A ring and a disc of the same radius and mass have different MI. The ring has more MI because all its mass is farther from the centre.
- A rod rotated about its end has a larger MI than when rotated about its centre.
- An ice skater pulling arms inward reduces the distance of mass from the axis, lowering MI and increasing spin rate.
4. Moment of Inertia of Common Shapes
Certain standard shapes have well-known MI values. These results are used frequently in physics and engineering.
4.1. Standard MI Values
- Uniform Rod (axis through centre): \( I = \dfrac{1}{12} ML^2 \)
- Uniform Rod (axis through one end): \( I = \dfrac{1}{3} ML^2 \)
- Solid Disc (axis through centre): \( I = \dfrac{1}{2} MR^2 \)
- Solid Sphere: \( I = \dfrac{2}{5} MR^2 \)
- Hollow Ring: \( I = MR^2 \)
5. Physical Meaning of Moment of Inertia
Moment of inertia tells us how much torque is needed to change the rotational motion of a body. A higher MI means the body is harder to rotate.
5.1. Relation with Angular Acceleration
Just as Newton’s second law relates force and acceleration, rotational motion follows:
\( \tau = I \alpha \)
Here:
- \( \tau \) is torque
- \( \alpha \) is angular acceleration
This shows that for the same torque, a body with larger MI rotates more slowly.
6. Why Moment of Inertia Matters
Moment of inertia plays a key role in mechanical design, sports, robotics, vehicle dynamics, and rotational stability. Understanding MI allows us to predict how rotating systems behave and how much torque is required to control them.
In the next topics, we build upon MI to study radius of gyration, theorems of MI, and complete rotational dynamics.