Derivation of moment inertia formula

In summary, the formula for moment of inertia of a uniform thin rod of length l about an axis through its center perpendicular to the rod is 1/12 Ml^2, where M is the mass of the rod and l is the length. This can be derived by using the formula \int R^2.dm and integrating from -l/2 to l/2, with dm being equal to p.A.dx, where p is the density and A is the cross-sectional area.
  • #1
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How do I derive the formula 1/12 Ml^2?
Derive the formula for moment of inertia of a uniform thin rod of length l about an axis through its center perpendicular to the rod.
 
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  • #2
There are a few ways to do it. Moment of inertia is calculated by
[tex]\int R^2.dm[/tex]

So place x=0 at the centre, the x-axis running along the rod. So you're integrating from -l/2 to l/2.
We must find dm in terms of our integration variable x. In dx we have an element of mass dm.
mass = (density)(volume)=(density)(cross-sectional area)(length)

So
dm = p.A.dx
where p is the density and A the cross-sectional area. Our integral is now:
[tex]\int_{-l/2}^{l/2} pAx^2.dx[/tex]
If you work it out you find it equals:
[tex]\frac{1}{12} pAl^3[/tex]

but if we remember that mass = pAl, then we get 1/12 Ml^2.
 
  • #3


The moment of inertia is a measure of an object's resistance to rotational motion. It depends on the mass distribution of the object and the distance of the mass from the axis of rotation. In the case of a thin rod of length l, the mass is distributed evenly along its length and the axis of rotation is through its center.

To derive the formula for moment of inertia, we will use the basic definition of moment of inertia, which is the sum of the products of mass and square of distance from the axis of rotation. Mathematically, it can be written as:

I = Σm*r^2

Where I is the moment of inertia, m is the mass of a small element of the rod, and r is the distance of that element from the axis of rotation.

To find the moment of inertia of the entire rod, we need to integrate this expression over the entire length of the rod. Since the mass is evenly distributed, we can also express it as the product of mass per unit length (M/l) and the length (l).

Therefore, the moment of inertia of the entire rod can be written as:

I = ∫(M/l)*r^2*dr

Where the integral is taken from 0 to l, representing the entire length of the rod.

Now, we can solve this integral to get the final formula for moment of inertia:

I = (M/l)*∫r^2*dr
= (M/l)*[(r^3)/3] from 0 to l
= (M/l)*[(l^3)/3 - (0^3)/3]
= (M*l^2)/3

Finally, we can rearrange this formula to get the commonly used formula for moment of inertia of a thin rod:

I = (1/12)*M*l^2

This formula shows that the moment of inertia of a thin rod is directly proportional to its mass and the square of its length, with a constant value of 1/12. This means that a longer and heavier rod will have a higher moment of inertia, making it more difficult to rotate.

In conclusion, the derivation of the moment of inertia formula for a thin rod involves using the basic definition of moment of inertia and integrating it over the entire length of the rod. This formula is applicable to other uniform objects as well, as long as their mass is evenly distributed and the axis of rotation is through their center.
 

1. What is the moment of inertia formula?

The moment of inertia formula, also known as the second moment of area formula, is used to calculate the distribution of mass around an axis of rotation. It is represented by the symbol I and is given by the integral of r^2 dm, where r is the distance from the axis of rotation and dm is the infinitesimal mass element.

2. How is the moment of inertia formula derived?

The moment of inertia formula can be derived using the basic principles of calculus and the definition of moment of inertia. It involves breaking down an object into infinitesimal mass elements, calculating the moment of inertia for each element, and then integrating them to find the total moment of inertia for the entire object.

3. What are the units of the moment of inertia formula?

The units of the moment of inertia formula depend on the units of the mass and distance used. In SI units, the moment of inertia is measured in kilograms per square meter (kg/m^2) or in meters squared (m^2). In US customary units, it is measured in slugs per square foot (slug/ft^2) or in feet squared (ft^2).

4. How is the moment of inertia formula used in real-life applications?

The moment of inertia formula is used in a variety of real-life applications, including the design of bridges, buildings, and other structures to ensure their stability and strength. It is also used in the study of rotational motion and dynamics in physics and engineering, and in the design of rotating machinery such as turbines and motors.

5. Are there any limitations to the moment of inertia formula?

While the moment of inertia formula is a useful tool for calculating the distribution of mass around an axis of rotation, it does have some limitations. It assumes that the object is rigid and has a constant density, which may not always be the case in real-world scenarios. In addition, it may not accurately represent the moment of inertia for complex or irregularly shaped objects.

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