Theorems of perpendicular and parallel Axis

Before studying perpendicular and parallel axis theorem in class 11, we must understand the concept of moment of inertia. Moment of inertia is nothing but the tendency that prevents a body from following an angular acceleration. Its mathematical formula suggests that the moment of inertia is equal to the sum of the mass product of the moving body’s each particle and the square of its distance from the rotational axis. 

The same body can have different moments of inertia according to the axis at which it is rotating. For example, one can rotate a rod easily by holding its centre but a higher force or torque will be required to rotate a rod by holding its ends. 

In rotational dynamics, the mass of an object is equal to its moment of inertia. It is a well known fact that lighter objects can be moved easily as compared to the heavier objects. Likewise, objects with less moment of inertia can be rotated easily as compared to the objects with greater moment of inertia. 

Therefore, Moment of inertia

I = ∑miri2

Here mi is the mass of ith object and ri is position of  ith object.

The theorems of perpendicular and parallel axes are taught to facilitate easy calculation of moment of inertia. 

State Parallel and Perpendicular Axis Theorem 

Perpendicular Axis Theorem 

Perpendicular Axis Theorem applies to flat bodies that have a minimum thickness i.e. planar bodies. According to the perpendicular axis theorem, when a planer of flat object’s moment of inertia is measured, it becomes equal to the addition of the moments of inertia of 2 perpendicular axes that meet the perpendicular axis at the same point (concurrent). Also, these perpendicular axes should lie in the same plane in which the planer body is. 

Therefore, according to the perpendicular axis theorem, the formula to calculate the moment of inertia is given below:

Iz = Ix + Iy

Here Ix and Iy are moments of inertia along the x-axis and y-axis respectively.

Parallel Axis Theorem 

Parallel axis theorem in class 11 is true for bodies of any size and shape. According to this theorem, the calculation of the an object’s moment of inertia at an axis which passes through its centre of mass becomes equal to the addition of an object’s moment of inertia going through its centre of mass and the product of the mass of that object and square of the length that lies between the two axes. 

Therefore, according to the parallel axis theorem in class 11, the formula to calculate the moment of inertia is given below:

I = Ic +Mh2

Here, 

h is the distance or length between both the axes

M is the object’s mass 

I is the object;s moment of inertia 

Ic is the moment of inertia at the centre 

Radius of Gyration 

The distance from the rotational axis to an imaginary point at which the entire mass of the body is considered to be concentrated is called the radius of gyration. Therefore, if the particles of an object are located near to the rotational axis, the radius of gyration is relatively less. However, if the particles of the object are located far away from the rotational axis, the radius of gyration is relatively more. 

A body’s moment of gyration can sometimes be shown by utilising the radius of gyration. It is often denoted by ‘k’. 

Mathematically, radius of gyration can be expressed as below:

I = mk²

Where m is the object’s mass and I is its moment of inertia. 

Therefore, as per this formula the value of k comes to be,

k=√I÷m

Uniform Disc’s MOI at a tangent perpendicular to its plane

A uniform disc’s moment of inertia at a transverse axis going through its centre can be expressed as follows:

I = MR2÷2=IG

To find the moment of inertia at a tangent that is perpendicular to its plane, we must remember that these two axes run parallel to each other. Therefore, as per the parallel axis theorem we have, 

By using the above expression, we can easily find a uniform disc’s MOI at its tangent that is perpendicular to the plane in which the disc lies.

To find the moment of inertia at the diameter, consider three axes in a way that x and y axes fall in the disc’s plane and the z axis follows the disc’s geometrical axis. G i.e. the centre of mass of the object is the origin of the system. 

Now we have, 

Applications of Radius of Gyration 

The comparison between the behaviour of objects of different shapes under compression can be studied by using the radius of gyration. 

Applications of Perpendicular and Parallel Axis 

• The MOI of a body that is rotating can be found by using perpendicular and parallel axis theorem, therefore, they can be utilised in several engineering applications

• The parallel axis theorem can be used to find the moment of inertia when any of the parallel axes passes through the centre of mass of an object

• Radius of gyration can be measured with the help of the perpendicular axis theorem

Conclusion 

The above sections explain parallel axis theorem and perpendicular axis theorem and related concepts. According to the perpendicular axis theorem, when a plane of a flat object’s moment of inertia is measured, it becomes equal to the addition of the moments of inertia of 2 perpendicular axes that meet the perpendicular axis at the same point. According to parallel axis theorem, the calculation of the an object’s moment of inertia at an axis which passes through its centre of mass becomes equal to the addition of an object’s moment of inertia going through its centre of mass and the product of the mass of that object and square of the length that lies between the two axes.

• Perpendicular and parallel axis theorem in class 11 together i.e. in conjunction are useful for determining a rigid object’s moment of inertia at any axis.