Moment of Inertia: Calculation Method

In summary, the ratio of moment of inertia of a circle and that of a square having the same area about their centroidal axis is equal to 3/π. This calculation assumes that the centroidal axis is in the plane, and not normal to it.
  • #1
Benjamin_harsh
211
5
Homework Statement
Ratio of moment of inertia of a circle and that of a square having same area about their centroidal axis is:
Relevant Equations
Why they calculated ##\large\frac{d}{a}## in first step?
Ratio of moment of inertia of a circle and that of a square having same area about their centroidal axis is:

Sol: both area and square have same area:

##a^2 = \large\frac{π}{4}\normalsize d^{2}; \large\frac{d}{a} =\large \sqrt\frac{4}{p}##

Ratio of moment of inertia of a circle and that of a square about their centroidal axis is:

##\large\frac{I_{c}}{I_{s}} = \large\frac{\frac{π}{64}d^4}{\frac{a^{4}}{12}} = \large\frac{12π}{64}\frac{d}{a}^{4}##

##\large\frac{3π}{16}(\sqrt\frac{4}{p})^{4} = \frac{3}{π}##
 
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  • #2
It could have been calculated later, but that ratio is needed for the equal areas
is
 
  • #3
Benjamin_harsh said:
moment of inertia of a circle and that of a square having same area about their centroidal axis
These questions you are working through keep referring, ambiguously, to "the" centroidal axis. I have tended to assume they mean the one normal to the plane. In the present question it doesn’t matter whether it is that axis or any axis in the plane as long as the choice is consistent, but looking at the detailed calculation they seem to be referring to an axis in the plane. If they meant normal to the plane then both values would be doubled.
Interesting...
 
  • #4
Correction in last step: ##\large\frac{3π}{16}(\sqrt\frac{4}{π})^{4} = \large \frac{3}{π}##.
 
Last edited:

1. What is moment of inertia?

Moment of inertia is a physical property of a rigid body that determines its resistance to angular acceleration about an axis of rotation. It is also known as rotational inertia.

2. How is moment of inertia calculated?

The moment of inertia of a rigid body can be calculated by using the mass distribution of the body and its distance from the axis of rotation. It is calculated by taking the sum of the products of each mass element and the square of its distance from the axis of rotation.

3. What is the formula for moment of inertia?

The formula for moment of inertia is I = Σmr², where I is the moment of inertia, m is the mass of each element, and r is the distance of each element from the axis of rotation.

4. What is the unit of measurement for moment of inertia?

The unit of measurement for moment of inertia depends on the unit of measurement used for mass and distance. In the SI system, the unit of measurement for moment of inertia is kilogram-meter squared (kg·m²).

5. Why is moment of inertia important?

Moment of inertia is important because it helps us understand how different objects will behave when subjected to rotational motion. It is also used in various engineering and physics applications, such as calculating the stability of structures and predicting the motion of rotating objects.

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