# Moment of inertia with system of rods

• symbolic
In summary: That makes no sense to me. I am still unfamiliar with this particular shape and don't know if that is possible. I am assuming that the rods are identical and M is the mass of a single rod. This is why I asked for a diagram.Hopefully someone else can help you more. It seems like this is fairly advanced physics and I am not sure if you already should know this or not. If you don't, then you probably need to go back and cover rotational inertia more thoroughly. If you do, then you should be able to apply your existing knowledge to this problem.In summary, the problem involves finding the moment of inertia for a system of three slim rods placed in an inverted U shape
symbolic

## Homework Statement

Three slim rods with the same length L are placed in a inverted U shape, as the figure shows. The masses of the vertical rods are the same, while the third bar has the mass 3 times bigger than the first or the second rod. The rods' thickness should be ignored. Find the moment of inercia of the system in relation to the following axes:

(a) containing each bar;
(b) parallel to the plane of the page and perpendicular to the bars, going through their centers of mass;
(c) perpendicular to the plane of the page, going through the center of mass of each bar;
(d) perpendicular to the plane of the page, going through the center of mass of the system xCM = L / 2 and yCM = 4 L / 5 taking the origin as the lowest point in the left vertical bar;

Answers: (a) 2 ML2, (2/3) ML2 e 2 ML2 ; (b) (11/12) ML2 e (3/4) ML2; (c) (35/12) ML2 , (17/12) ML2 e (35/12) ML2; (d) (73/60) ML2.

(question was translated, let me know if something is difficult to understand)

## The Attempt at a Solution

I tried to solve question (a) considering each bar as a point, that failed however.

Calculating Icm of the first bar with an axis going through it

Icm = M(0²) + 3M(L/2)² + M(M(0²)
Icm = 3/4ML²

I tried a few variations of this but couldn't reach the answer 2 ML2, so now I'm stuck. Any help is appreciated.

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For some reason it gives me an errror when I try to edit the topic. Here are the correct answers:

(a) 2 ML², (2/3) ML² e 2 ML² ; (b) (11/12) ML² e (3/4) ML²; (c) (35/12) ML² , (17/12) ML² e (35/12) ML²; (d) (73/60) ML².

and M(M(0²) should be M(0²)

Your discrete definition of rotational inertia also has a continuous form which should be used for continuous bodies, like rods.

The inertia of the middle bar about one of the edge bars comes from that integral over all the mass. The inertia is not the same if you replace the body with a point-mass at its center, which is what your math seems to be doing.

Your math also seems to say that the inertia of one edge bar about the other edge bar is zero?

## 1. What is the definition of moment of inertia?

Moment of inertia is the measure of an object's resistance to rotational motion about a given axis. It is also known as rotational inertia.

## 2. What is the formula for calculating moment of inertia?

The formula for moment of inertia depends on the shape and mass distribution of the object. For a system of rods, the formula is I = Σmr², where I is the moment of inertia, m is the mass of each rod, and r is the distance from the axis of rotation to the center of mass of each rod.

## 3. How does the distribution of mass affect the moment of inertia?

The distribution of mass affects the moment of inertia in two ways. First, the farther the mass is from the axis of rotation, the greater the moment of inertia will be. Second, the shape of the object also plays a role in determining the moment of inertia, with objects with more mass concentrated at the edges having a higher moment of inertia.

## 4. How does the moment of inertia differ from mass?

The moment of inertia and mass are two different properties of an object. Mass is a measure of the amount of matter in an object, while moment of inertia is a measure of the object's resistance to rotational motion. In other words, an object with a large mass may not necessarily have a high moment of inertia, and vice versa.

## 5. How does moment of inertia play a role in rotational motion?

Moment of inertia is an important factor in rotational motion because it determines how much torque is needed to produce a given angular acceleration. Objects with a higher moment of inertia will require more torque to rotate at the same rate as objects with a lower moment of inertia.

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