# Rotational Inertia (Moment of Inertia) of a Rod

• tseryan
In summary, the problem involves finding the moment of inertia of a system with three masses placed on a light rod along the x axis. The solution involves finding the sum of the point masses of inertia and using the parallel axis theorem to find the moment of inertia about any parallel axis, including one end of the rod.
tseryan

## 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!

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.)

## What is rotational inertia?

Rotational inertia, also known as moment of inertia, is a measure of an object's resistance to changes in its rotational motion. It is dependent on the object's mass and distribution of mass relative to its axis of rotation.

## How is rotational inertia calculated?

The rotational inertia of a rod is calculated by multiplying its mass by the square of its distance from the axis of rotation. This can be represented by the formula I = m * r^2, where I is the moment of inertia, m is the mass, and r is the distance from the axis of rotation.

## What factors affect the rotational inertia of a rod?

The rotational inertia of a rod is primarily affected by its mass and distribution of mass. Objects with larger mass or mass located farther from the axis of rotation will have a greater moment of inertia.

## How does the shape of a rod affect its rotational inertia?

The shape of a rod can also affect its rotational inertia. Objects with a more compact shape, such as a solid cylinder, will have a higher moment of inertia compared to objects with a more spread out shape, such as a hollow cylinder.

## Why is understanding rotational inertia important in physics?

Understanding rotational inertia is important in physics because it helps us predict and analyze the motion of objects. It also plays a crucial role in the laws of conservation of angular momentum, which states that the total angular momentum of a system remains constant in the absence of external torques.

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