Moment of Inertia
- Consider trying to spin a bicycle wheel.
- If the mass is concentrated near the rim, it feels harder to start or stop the rotation compared to when the mass is closer to the center.
This resistance to rotational motion is described by the moment of inertia ($I$), which is the rotational equivalent of mass.
Definition of Moment of Inertia
Definition
Inertia
Inertia is the tendency of an object to resist changes in its state of motion.
In other words, it quantifies how difficult it is to change the rotational motion of an object.
It depends on two factors:
- Mass of the object
- Distribution of mass relative to the axis of rotation
Mathematically, the moment of inertia for a system of point masses is given by:
$$
I = \sum m_i r_i^2
$$
where:
- $m_i$ is the mass of each point
- $r_i$ is the distance of the mass from the axis of rotation
- Single particle: A particle of mass $m$ at a distance $R$ from the axis has a moment of inertia $I = mR^2$.
- Multiple particles: Two particles, each of mass $m$, at distances $R_1$ and $R_2$ from the axis have a total moment of inertia $I = mR_1^2 + mR_2^2$.
Moment of Inertia as the Rotational Equivalent of Mass
- In linear motion, mass measures an object's resistance to changes in velocity.
- Similarly, in rotational motion, the moment of inertia measures an object's resistance to changes in angular velocity.
- The moment of inertia depends on the axis of rotation.
- The same object can have different moments of inertia if rotated about different axes.
Rotational Kinetic Energy
Just as a moving object has kinetic energy, a rotating object has rotational kinetic energy.
Definition
Rigid body
A rigid body is an idealized object in which the relative positions of all particles remain fixed, regardless of external forces or torques.
Formula for Rotational Kinetic Energy
Definition
Rotational kinetic energy
Rotational kinetic energy or angular kinetic energy is kinetic energy due to the rotation of an object and is part of its total kinetic energy.
The rotational kinetic energy of a rigid body is given by:
