Moment of Inertia
Moment of inertia, also known as rotational inertia or angular mass, is a physical quantity that resists a rigid body’s rotational motion. It is analogous to mass in translational motion. It determines the torque required to rotate an object by a given angular acceleration. Moment of inertia does not restrict itself to a rigid body only. It also applies to a system of particles rotating about a common axis.
How to Calculate Moment of Inertia
Point Mass
For a point mass (single body), the moment of inertia formula is given by the product of mass and the square of the object’s perpendicular distance from the axis of rotation.
Where
I is the moment of inertia
m is the mass
r is the perpendicular distance from the axis of rotation
System of Particles
Suppose n particles of masses m_{1}, m_{2}, …, m_{n} rotate about a common axis at perpendicular distances r_{1}, r_{2}, …, r_{n} from the axis. Then the moment of inertia of the system is
The moment of inertia of a system of particles is the sum of the moments of inertia of the individual particles taken about a common axis.
Units and Dimensions
The SI unit of moment of inertia is kgˑm^{2}, and the cgs unit is gˑcm^{2}. The dimension is [M L^{2 }T^{0}].
Integral Moment of Inertia
For a continuous mass distribution, the integral form of moment of inertia is given by
Where dm is the instantaneous mass.
The above equation can also be written in terms of density ρ and instantaneous volume dV as follows.
Moment of Inertia Table
Several everyday objects, such as rotating disks, cylinders, and spheres, have welldefined moment of inertia formulas. A chart consisting of the different formulas is listed in the table below. Here M represents mass, R represents radius, and L represents the length.
Object  Moment of Inertia Formula  Object  Moment of Inertia Formula 

Solid Sphere  (2/5)MR^{2}  Rectangular plate with sides of length a and breath b and axis passing perpendicularly through the center  (1/12)M(a^{2} + b^{2}) 
Hollow ThinWalled Sphere  (2/3)MR^{2}  Rectangular plate with sides of length a and breath b and axis passing perpendicularly through the edge  (1/3)Ma^{2} 
Solid Cylinder  (1/2)MR^{2}  Rectangular plate with sides of length a and breath b and axis passing parallelly through the center  (1/12)Ma^{2} 
Hollow ThinWalled Cylinder  MR^{2}  Slender rod with the axis through the center  (1/12)ML^{2} 
Hollow Cylinder, with inner radius R_{1} and outer radius R_{2}  (1/2)M(R_{1}^{2} + R_{2}^{2})  Slender rod with the axis through one end  (1/3)ML^{2} 
Moment of Inertia and Rotational Kinematics
The moment of inertia of a rotating object about a fixed axis is useful in calculating a few key quantities in rotational motion. Newton’s second law for rotation gives a relationship between torque, moment of inertia, and angular acceleration. According to this law,
Where
τ is the applied torque
I is the moment of inertia
α is the angular acceleration
The relationship between moment of inertia and rotational kinetic energy is given by
And with angular momentum, the equation is
Where
K is the rotational kinetic energy
L is the angular momentum
ω is the angular velocity
Factors Affecting Moment of Inertia
The moment of inertia depends upon
 The shape of the object
 Mass distribution or density
 Location of the axis of rotation

References
Article was last reviewed on Wednesday, August 2, 2023