Moment the inertia that a rod who axis goes v the centre of the rod, having actually mass (M) and also length (L) is generally expressed as;


The minute of inertia can also be to express using an additional formula once the axis the the stick goes v the end of the rod. In this case, us use;


Moment that Inertia Of stick Formula derivation / Calculation

*

Let us understand the source of the moment of inertia because that the two moments.

You are watching: Moment of inertia of a thin rod

1. Once the axis is through the center of the mass.

We have to keep in psychic a few things;

We need to imagine that the pole is divided into countless pieces that infinitesimally slim slices.Each slice will have actually a size of dx and a mass of dm.We need to identify the variable to it is in summed.

In the an initial instance, the rod must be take away to have an infinite number of point masses. Every one of their products will be derived by multiplying the square the the distance from the axis. We additionally have to consider the exceedingly tiny element of size dl corresponds to the massive dm. Besides, once we picture or brand the origin at the centre of mass the is lying on the line of the axis, us will view that distance of the rod indigenous the left is -L/2, when the street from the origin to its right is +L/2.

Now,

If us look in ~ the pole we have the right to assume the it is uniform. Because of this the linear thickness will remain constant and we have;

or = M / together = dm / dl

dm = (M / L) dl

Now we need to replace the worth of dm in the expression;

*

I = O ∫M r2dm

I = -L/2 ∫L/2 l2 (M/L)dl

Here the change of the integration is the size (dl). The boundaries have readjusted from M to the required fraction of L.

I = -L/2 ∫L/2 (M/L)l2dl

I = M / together -L/2L/2 I = M / 3L <(L / 2)3 – (-L / 2)3>I = M / 3L I = (M / 3L) . 2L3 / 8

I = (1 / 12) ML2

2. When the axis is with the finish of the rod.

*

For calculating as soon as the axis is in ~ the finish we have actually to attract the beginning at that certain end.

The very same expression have the right to be used but with one more limit. Because the axis rests in ~ the end, the limit the is provided in integration is 0.

See more: ~ How Many Florets In A Head Of Cauliflower ? Weight Equivalents: Cauliflower

I = O∫M r2dm

I = O∫L l2 (M / L) dl

If we use the integration;

Iend = (M / l )O∫L l2 dl

Iend = (M / L) 0LIend = (M / 3L) < L3 – 03>Iend = (⅓) ML2

Alternatively, the parallel axis to organize can additionally be offered to identify the expression.