Rotational Inertia (Moment of Inertia) of a Rod

AI Thread Summary
The discussion focuses on calculating the moment of inertia for a system of point masses attached to a massless rod. The key point is that the rotational inertia depends on the distribution of mass rather than just the center of mass. The correct approach involves using the formula I = MR^2 for each point mass and summing their contributions to find the total moment of inertia. The Parallel Axis Theorem is also relevant, as it allows for the calculation of rotational inertia about different axes. Understanding these principles is essential for solving problems related to rotational dynamics effectively.
tseryan
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Homework Statement


A very light (meaning don't consider mass of the rod) rod is placed along the x axis. It has a mass m1=2.0kg at x=0, a mass m2=1.50kg at x=50cm, and a mass m3=3.0kg at x=100cm.

Find the moment of inertia of the system about a pivot point at x=0.

Homework Equations



I=(1/3)M(L^2) -- Parallel Axis Theorem

I= integral of[(r^2)dm]?

The Attempt at a Solution



Because the mass of the rod does not matter, I'm thinking of finding the center of mass between the three masses and treating that as one whole mass at a certain point x. Does that idea make any sense? Any help would be greatly appreciated!
 
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That would be a bad idea. The rotational properties depend on the distribution of mass, not just the center of mass. Hint: What's the rotational inertia of a point mass about some axis?
 
It is I=MR^2. If I find the sum of the point masses of inertia I get the moment of inertia? Does it matter that it's on a rod?
 
tseryan said:
It is I=MR^2.
Right.
If I find the sum of the point masses of inertia I get the moment of inertia?
Absolutely.
Does it matter that it's on a rod?
Nope. (You need something to connect the masses as a rigid structure.)
 
Got it! Thanks Doc Al! Now I don't understand why there is a Parallel Axis Theorem for a rod with the equation I=(1/3)M(L^2). What would that be used for?
 
Not sure I understand your question. The parallel axis theorem applies to any object, not just a rod.

Starting with the rotational inertia of a rod about its center of mass (what's the formula for that?), the parallel axis theorem will allow you to find the rotational inertia of the rod about any other parallel axis--including about one end, which is what (1/3)M(L^2) is for. (Try it--it's easy.)
 

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