Rotational Motion
Moment of Inertia
Let us take a particle at a distance from the axis of rotation. Linear velocity of this particle is as follows:
`v_i=r_iω`
So, kinetic energy of this particle can be given as follows:
`k_i=1/2m_iv_i^2``=1/2m_ir_i^2ω^2`
Total kinetic energy K of the body can be given by the sum of KE of individual particles.
`K=Σk_i`
`=1/2Σ(m_ir_i^2ω^2)`
As ω is same for all particles, we take out ω from the sum and we get following equation.
`K=1/2ω^2(Σm_ir_i^2)`
Here, we get a new character of a rigid body, and it is called moment of inertia I. Moment of inertia is given by following equation.
`I=Σm_ir_i^2`
The moment of inertia is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis. It is also known as the mass moment of inertia, angular mass or rotational inertia.
Now, equation for total kinetic energy can be expressed as follows:
`K=1/2Iω^2`
Moment of inertia is independent of the magnitude of angular velocity. It is a characteristic of the rigid body and the axis about which it rotates.
Unlike the mass of the body, the moment of inertia is not a fixed quantity. It depends on distribution of mass about the axis of rotation, and the orientation and position of the axis of rotation with respect to the body as a whole.
Radius of Gyration
The distance from the axis of a mass point (whose mass is equal to the mass of the whole body and whose moment of inertia is equal to the moment of inertia of the body about the axis) is called the radius of gyration.
Theorem of Perpendicular Axes
The moment of inertia of a planar body (lamina) about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two perpendicular axes concurrent with perpendicular axis and lying in the plane of the body.
This figure shows a planar body. Here, z-axis is taken as the perpendicular axis through a point O. Two mutually perpendicular axes lying in the plane of the body and concurrent with z-axis are x- and y-axes. According to the theorem:
`I_z=I_x+I_y`
Theorem of Parallel Axes
The moment of inertia of a body about any axis is equal to the sum of the moment of inertia of the body about a parallel axis passing through its center of mass and the product of its mass and square of the distance between the two parallel axes.
In this figure, z and z' are two parallel axes, and the gap between them is a. The z-axis passes through the center of mass O of the rigid body. According to the theorem of parallel axes:
`I_(z')=I_z+Ma^2`