Parallel and perpendicular axes theorems

What is meant by Parallel Axis Theorem?

The parallel axis theorem can be defined as a body’s moment of inertia about an axis parallel to its middle is equal to the mass of the body times the square of the distance between the two axes and the moment of inertia about the axis going through the middle.

What is meant by the Perpendicular Axis Theorem?

According to the perpendicular axis theorem, the moment of inertia about any axis in a plane body, which are perpendicular to the plane, is equal to that of any two perpendicular axes in the plane that cross the plane’s main axis.

Parallel Axis Theorem Formula

Parallel axis theorem articulation can be communicated as follows:

I = Ic + Mh2
• I resembles the moment of inertia of the body

• Ic resembles the moment of inertia about the centre

• M resembles the mass of the body

• h2 resembles the square of the distance between the two axes

Parallel Axis Theorem Derivation

Allow Ic to be the moment of inertia of an axis that is going through the focal point of mass (AB from the figure) and I is the moment of inertia about the axis A’B’  present at a distance of h.

Think about a molecule of mass m at a distance r from the centre of gravity of the body.

So,

Distance from A’B’ = r + h

I = ∑m (r + h)2

I = ∑m (r2 + h2 + 2rh)

I = ∑mr2 + ∑mh2 + ∑2rh

I = Ic + h2∑m + 2h∑mr

I = Ic + Mh2 + 0

I = Ic + Mh2

The formula of the parallel axis theorem is derived above.

Parallel Axis Theorem of Rod

The parallel axis theorem of the rod is still up in the air by tracking down the moment of inertia of the rod.

Moment of inertia of rod is written as:-:

I = 1/3 ML2

The distance between the finish of the rod and its middle is given as:

h = L/2

so, the parallel axis theorem of the rod is:

Ic = 1/3 ML2 – ML/2*2

Ic = 1/3 ML2 – 1/4ML2

Ic = 1/12 ML2

Perpendicular Axis Theorem Formula

In the event that the moment of inertia around two of the axes are known the moment of inertia about the third axis can be tracked down utilising the articulation:

Ia=Ib+Ic

Say in a designing application we need to track down the moment of inertia of a body, however the body is irregularly shaped, and the moment of in these cases we can utilise the parallel axis theorem to get the moment of inertia anytime as long as we probably are aware the focal point of gravity of the body. Perpendicular axis theorem is utilised when the body is symmetric in shape around two out of the three axis.This is an exceptionally valuable theorem in space material science where the computation of moment of inertia of space-crafts and satellites, making it workable for us to arrive at the external planets and, surprisingly, the deep space. 

Use of  parallel axis theorem

Using the parallel axis theorem, we can observe the moment of inertia of the region of an inflexible body whose axis is parallel to the axis of the realised moment body, and this occurs through the focal point of gravity.

Applications of perpendicular axis theorem

The applications of perpendicular axis theorem are

• A three-dimensional object’s moment of inertia can be determined by the perpendicular axis theorem.
• By the perpendicular axis theorem, we can calculate the moment of inertia around the third axis.

Conclusion

Area moments of inertia about its centroid of the ordinary areas are accessible in standard tables. Yet, commonly you will come to the circumstance where you might need to calculate the moment of inertia about some other axis not going through the centroid. With regards to its centroid, the customary areas are accessible in standard tables. In any case, ordinarily you will come to the circumstance where you might need to calculate the area moment of inertia about some other axis not going through the centroid. The parallel axis theorem and the perpendicular axis theorem are helpful for calculating the area moment of inertia of such cases.