# Calculating Moment of Inertia of Particles & Cubes

• joemama69
In summary, the conversation discussed calculating the moment of inertia for a construction of four point particles arranged in a square and a cube with a sphere cut out of it, both pivoted about an axis perpendicular to their centers. The moment of inertia for the square construction was calculated to be 8ma^2, while the moment of inertia for the cube with the cut-out sphere was calculated to be 17/120 ma^2.
joemama69

## Homework Statement

a) Four point particles, each of mass m, are arranged in a square and attached by massless rods that run along the sides of the square, and along the diagonals. The length of each side of the square is a. The construction is then pivoted about an axis perpendicular to the square, and through its center. Calculate moment of inertia Ia about this axis.

b) pose a sphere of radius a/4 is cut out of a cube. The center of the excised sphere is at the center of the cube. What is the moment of inertia Id of the resulting object, pivoted about an axis perpendicular to one of the sides and through the center of the cube?

## The Attempt at a Solution

part a

I = $$\sum$$mr2 where r = $$\sqrt{2a^2}$$

I = 4(mr2) = 8ma2

part b

Icube = 1/12 m(2l2)...Isphere = 2/5 mr2

I = Ic - Is = 1/12 m(2a2) - 2/5 mr2 = 17/120 ma2

#### Attachments

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• 2b.pdf
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Do you just want confirmation, or are you unsure about what you did? Both answers are correct.

both i suppose, thanks

## 1. What is moment of inertia?

Moment of inertia is a measure of an object's resistance to rotational motion. It is the sum of the products of each particle's mass and its distance from the axis of rotation squared.

## 2. How do you calculate moment of inertia for particles?

The moment of inertia for a single particle can be calculated using the formula I = mr^2, where m is the mass of the particle and r is the distance from the axis of rotation.

## 3. How do you calculate moment of inertia for a cube?

The moment of inertia for a cube can be calculated by dividing the cube into smaller particles and using the parallel axis theorem. This involves calculating the moment of inertia for each particle, and then adding them together while taking into account the distance between the particles and the axis of rotation.

## 4. What is the parallel axis theorem?

The parallel axis theorem states that the moment of inertia of an object is equal to the moment of inertia of the object's center of mass plus the product of the object's mass and the square of the distance between the center of mass and the new axis of rotation.

## 5. Why is calculating moment of inertia important?

Calculating moment of inertia is important in understanding an object's rotational motion and its stability. It also helps in determining the amount of torque needed to cause an object to rotate, and can be used to analyze complex systems such as rotating bodies or machinery.

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