# Moment of inertia of rods

Ok so I'm extremely comfortable with calculating moment of inertia of continuous bodies but how do we do it for a system not continuous. For example if 3 rods of mass $m$ and length $l$ are joined together to form an equilateral triangle what will be the moment of inertia about an axis passing through its centre of mass perpendicular to the plane. i know that moment of inertia of each rod is $ml^2/12$ and c.o.m is at centroid? also if 2 rods form a cross then to calculate the moment of inertia about its point of intersection would it be correct to sum up the individual moment of inertia of the rods form??

• Moments of inertia are additive. – CuriousOne Oct 10 '14 at 7:21

The moment of inertia for a system of $n$ point masses, $m_i$, at distances $r_i$ from the pivot is simply:

$$I = \sum m_i r_i^2 \tag{1}$$

We normally calculate $I$ by integration, i.e. we take each point mass to be an infinitesimal element of our continuous object and integrate to add up the moments of inertia of all those elements.

In your case let's call the three rods $A$, $B$ and $C$, then our initial equation (1) can be written as:

$$I = \sum m_{Ai} r_{Ai}^2 + \sum m_{Bi} r_{Bi}^2 + \sum m_{Ci} r_{Ci}^2$$

where all we've done is divide up our sum into the infinitesimal parts that belong to the three masses. But from equation (1) we know that $I_A = \sum m_{Ai} r_{Ai}^2$, and likewise for $B$ and $C$, so the total moment of inertia is just:

$$I = I_A + I_B + I_C$$

So just calculate the separate moments of inertia for all the objects in your system then add them together. In your particular case the objects are identical so the total is just the moment of inetria of a single rod multiplied by three.

• This will give us the moment of inertia about which axis? – Archer Oct 29 '17 at 16:56

The moment of inertia is defined relative to the point of rotation, which in this case is the centre of the equilateral triangle. Then you can multiply this result by 3.

@DrChuck's answer is correct. Generally speaking the total moment of inertia is the sum of the moments inertia calculated individually. You have to be careful about the the axis of rotation thought: if you wanted to calculate the moment of inertia (with respect to any axis) of a T shape created from 2 identical rods, you would calculate the moment of inertia of each rod differently because the axis of rotation as seen from the perspective of the rod would be different in each case.

Also if it's homework we can neglect the contact point of the different rods. In engineering we would have to alter the rods to make them stay together in a shape.
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}