Parallel Axis Theorem

In several cases, the moment of inertia about an axis which passes through the center of a shape is known (or calculated easily) and the moment of inertia of area on a second axis parallel to the first axis is needed. These two moments of inertia are combined by the parallel axes theorem.

Inertia

Inertia is the resistance of a physical body/object to a change in its velocity. This includes the changes to the object’s speed, or in the direction of motion of the body. One aspect of this property is the tendency of objects to move in a straight line at constant speed when no force is acting on them. The amount of inertia which is possessed by an object is proportional to its mass. But the inertia is not the same as the mass or momentum (product of the velocity and mass). The mass of an object may be measured by noticing the extent of its inertia. This is done by determining the amount of force needed to produce a certain acceleration.

Moment of Inertia

The moment of inertia is defined as the quantity expressed by the body resisting angular acceleration, which is the sum of the product of the mass of each particle and the square of it’s distance from the axis of rotation of an object. Or, more simply put, it can be described as a quantity that determines the amount of torque required for a given angular acceleration on an axis of rotation. The SI unit is kg.m².
The moment of inertia is
I = m x r²
Here,
m = Summation of product of the mass.
r = distance.
I = Moment of Inertia.

Law of Inertia

Law of inertia is the Newton’s first law of motion. According to the law of inertia if a body is at rest or moving at a constant velocity in a straight line then the body will remain at rest or keep moving in a straight line at the constant speed unless an external force acts on it. Before Galileo, it was assumed that all horizontal motion required a direct cause, but Galileo concluded from his experiments that a body in motion would remain in motion unless a force (like friction) would bring it to a standstill.

Parallel Axes Theorem

According to the parallel axes theorem the moment of inertia of a body with respect to an axis which is parallel to the body is the sum of the moment of inertia of a body with respect to the axis passing through the medium and the product of the mass of body and the square of distance between the two axes.

Parallel Axis Theorem Formula

Parallel axis theorem formula is given as
I = I_c + Mh²
Here,
I = moment of inertia of body
I_c= moment of inertia around centre
M = mass of body
h = distance between two axes

Parallel Axes Theorem Derivation

Let us consider Ic be the moment of inertia around the centre of an axis passing through the centre of mass and I is the moment of inertia of the body.
Let us consider the mass of a particle is m and r is the distance between the particle and the centre of gravity of the body.
Therefore, distance = r+h
I = ∑ m(r+h)²
I = ∑ m(r²+h²+2rh)
I = ∑ mr² +∑mh² +∑2rh
I= Ic  + h²∑m + 2h∑r
I= Ic  + h²m + 0
Therefore,
I= I_c  + mh² ———— (1)
Thus, equation one is the parallel axis theorem formula.

Perpendicular Axis Theorem

The perpendicular axis theorem only applies to flat or planar bodies. Flat bodies with very little or negligible thickness. According to this theorem, the moment of inertia of a plane body about an axis which is perpendicular to its plane is equal to the sum of its moments of inertia about two perpendicular axis that are coincident with the normal axis and lie in the plane of the body.
According to perpendicular axis theorem
I_x + I_y = I_z
Let us consider three axis, x axis, y axis, and z axis.
Moment of inertia of x axis is I_x = mx²
Moment of inertia of y axis is I_y = my²
Moment of inertia of z axis is I_z = m√(x² + y²)²
Therefore,
I_x + I_y = mx² + my²
I_x + I_y = m(x² + y²)
I_x + I_y = m√(x² + y²)²
Thus,
I_x + I_y = I_z
Therefore, the perpendicular axis theorem is proved.

Applications of Parallel and Perpendicular Axis Theorem

The calculation of moment of inertia is simplified by using perpendicular and parallel axis theorem together.
Perpendicular and Parallel axis theorem together help to study rotational dynamics of rigid bodies easily.

Centre of Gravity

The Centre of gravity is considered as the point through which the gravitational force acts on a body or system. Centre of gravity is the point about which the resultant torque which is caused by the gravitational force disappears. In cases where the gravitational field is considered to be uniform, the centre of gravity and the centre of mass is the same. Sometimes these two terms, centre of gravity and centre of mass, are used interchangeably because they are frequently said to be in the same position or location.

Conclusion

Inertia is the resistance of a physical body/object to a change in its velocity.
The amount of inertia which is possessed by an object is proportional to its mass.
The moment of inertia is
I = m x r²
Parallel axis theorem formula is given as
I= I_c  + mh²
According to perpendicular axis theorem I_x + I_y = I_z