Moment of Inertia of a Thin Uniform Rod

1. Concept Overview

The moment of inertia (M.I.) of a body depends on the distribution of its mass about the axis of rotation.
For a thin uniform rod, the M.I. varies depending on whether the axis passes through the center or one end of the rod.
A “thin” rod means its thickness is negligible compared to its length, so mass distribution is considered along one dimension.

2. Derivation

Let:

• Length of the rod = [ L ]
• Mass of the rod = [ M ]
• Linear mass density = [ \lambda = \dfrac{M}{L} ]

Consider an infinitesimal element of length [ dx ] at a distance [ x ] from one end.
The mass of this element is [ dm = \lambda , dx ].

(A) Moment of Inertia about an Axis through the Center and Perpendicular to the Length

For this case, the origin is at the center of the rod.
So, the limits are from [ -\dfrac{L}{2} ] to [ +\dfrac{L}{2} ].

[
I_{center} = \int_{-\dfrac{L}{2}}^{+\dfrac{L}{2}} x^2 dm = \lambda \int_{-\dfrac{L}{2}}^{+\dfrac{L}{2}} x^2 , dx
]

$ \displaystyle {{I}_{{center}}}=\lambda \left[ {\frac{{{{x}^{3}}}}{3}} \right]_{{-\frac{L}{2}}}^{{+\frac{L}{2}}}=\lambda \frac{{{{L}^{3}}}}{{12}}$

Substituting [ \lambda = \dfrac{M}{L} ]:

[
\boxed{I_{center} = \dfrac{1}{12} M L^2}
]

(B) Moment of Inertia about an Axis through One End and Perpendicular to the Length

Now, take the axis through one end of the rod.
The limits are from [ 0 ] to [ L ].

[
I_{end} = \int_{0}^{L} x^2 , dm = \lambda \int_{0}^{L} x^2 , dx
]

$ \displaystyle I_{end} = \lambda \left[ \dfrac{x^3}{3} \right]_0^L = \lambda \dfrac{L^3}{3}$

Substituting [ \lambda = \dfrac{M}{L} ]:

[
\boxed{I_{end} = \dfrac{1}{3} M L^2}
]

3. Relation between the Two Axes

Using the Parallel Axis Theorem:

[I_{end} = I_{center} + M \left(\dfrac{L}{2}\right)^2] [= \dfrac{1}{12} M L^2 + \dfrac{1}{4} M L^2 = \dfrac{1}{3} M L^2]

This verifies the derived results are consistent.

4. Physical Significance

• The M.I. about one end is four times that about the center.
• The farther the axis from the center, the greater the resistance to rotational motion.
• These results are fundamental for analyzing rotating systems like pendulums, rods, and beams.

5. Key Features

• Rod is uniform and thin, so only one dimension contributes to M.I.
• The M.I. depends on both mass and square of length.
• The results follow directly from the definition of M.I. and Parallel Axis Theorem.

6. Important Formulas to Remember

7. Conceptual Questions

8. FAQ / Common Misconceptions