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Introduction:
The moment of inertia plays an important role in rotational kinematics as in how mass plays an important role in linear kinetics. The common characteristic between them is they resist change. The moment of inertia solely depends on the axis on rotation and varies when this is changed. For a mass, the moment of inertia is denoted as mr2, where “r” is the radius of the chosen axis and “m” refers to the mass of the body.
The moment of inertia of an extended rigid body is just the sum of all the little pieces of mass multiplied by the square of their distances from the rotating axis. For a body with regular shape and homogenous density, they are calculated using dimension, shape, and total mass.
Christiaan Huygens, in his research, coined the term “ moment of inertia” while working on the oscillation of a compound pendulum.
Leonhard Euler coined the term “moment of inertia” in his 1765 book Theoria motus corporum solidorum seu rigidorum, and it is incorporated into Euler’s second law.
Definition of Moment of inertia:
The moment of inertia is the resistance to the change in angular acceleration of the body. It is the sum of the product of the mass of every particle of the body and the square of the distance at which the object is placed from the rotating axis.
In simpler words, it can be said it’s the quantity that decides the amount of torque required for acceleration.
Formula :
The general formula used to calculate the moment of inertia :
I = m × r2
Where m = the sum of the product of the object’s mass.
r = the distance of the object from the rotating axis.
The dimensional formula: M1 L2 T0.
The SI unit: kg m2
It is constant for a rigid body with a specific axis of rotation.
Factors affecting the moment of inertia:
A rotational axis is widely used to express the moment of inertia. The distribution of mass around a rotational axis is the main determinant.
Therefore, it depends on the following factors:
- The density of the body
- Size and shape of the body.
- Relative distribution of mass around the axis.
The rotating bodies can be classified into 2 categories:
- System of particles or discrete
- Rigid bodies or continuous.
Evaluation of Moment of Inertia of a System of Particles:
For a point mass, it is
I = mr2
For a system of particles, it is the summation of each particle that is,
I = m1 (r1)2 + m2 (r2)2 + m3 (r3)2 …..
I = ∑ mi + ri 2
[Where ri is the perpendicular distance between the axis and the i-th mass mi particle.]
Evaluation of Moment of inertia of rigid bodies
The moment of inertia of rigid bodies can be calculated by integrating. If the system of Rigid Bodies is divided into an infinite number of particles, then its Mass, ‘dm’ and distance of the mass from the axis of rotation is ‘r’. Now, the moment of Inertia becomes,
I = ∫ r2 dm
Moment of Inertia of different Rigid bodies
Rigid Bodies | Moment of Inertia |
the moment of inertia of a uniform circular disc | I = 3/2 ML2 |
Rod (from centre) | I = 1/12 ML2 |
Rod (from end) | I = ⅓ ML2 |
Solid cylinder | I = ⅓ MR2 |
Solid cylinder central diameter | I = ¼MR2 + 1/12 ML2 |
Thin spherical shell | I = ⅔ MR2 |
Hoop (from symmetry axis) | I = MR |
Hoop (from diameter) | I = 1/2MR2 |
Conclusion:
For a rigid body, the moment of inertia emerges as a physical parameter that integrates its shape and mass in momentum, kinetic energy, and Newton’s equations of motion.
It is applied in both linear and angular moments, but the appearance of the moment of inertia in planar and spatial movement differs significantly.
The moment of inertia in planar movement is defined by a single scalar quantity. However, in spatial movement, the same computations are given by a 3 × 3 matrix dimension of a moment of inertia.
The article highlights the definition of the moment of inertia, factors affecting it, the moment of inertia of a system of particles, and Moment of inertia of rigid bodies.
