# What is and how do I use the Parallel axis theorem

• JJRKnights

#### JJRKnights

I have in my problem, a ball sitting on a cylindrical rod that pivots at the bottom,
some guy in the explanation said parallel axis theorem and came up with moment of inertia of:
I = (mL^2)/3 + [2Mr^2/5 + M(L+r)^2]

where L is the length of the rod, m is the mass of the rod, M is the mass of the ball, r is the radius of the ball.

I assume you've covered moments of inertia, the parallel axis theorem then states that if you're rotating your object about an axis parallel to a second axis where you know the rotation moment of inertia then the new moment of inertia about your displaced axis is equal to the "normal" moment plus a term $md^2$ where m is the mass of the object and d is the distance between the axes (?) .

## 1. What is the Parallel axis theorem?

The Parallel axis theorem is a mathematical principle that states the moment of inertia of a rigid body about any axis is equal to the sum of its moment of inertia about a parallel axis through its center of mass and the product of its mass and the square of the distance between the two axes.

## 2. How is the Parallel axis theorem used in science?

The Parallel axis theorem is used to calculate the moment of inertia of a rigid body about an axis that is not passing through its center of mass. This is particularly useful in physics and engineering applications, such as calculating the rotational motion of objects.

## 3. What are the key assumptions of the Parallel axis theorem?

The Parallel axis theorem assumes that the body is rigid, the mass is distributed evenly, and the axis of rotation is perpendicular to the plane of motion.

## 4. Can the Parallel axis theorem be applied to 3D objects?

Yes, the Parallel axis theorem can be applied to 3D objects as long as the key assumptions are met. In this case, the moment of inertia would be calculated for each axis of rotation and then added together to get the total moment of inertia.

## 5. How is the Parallel axis theorem related to the moment of inertia?

The Parallel axis theorem provides a way to calculate the moment of inertia of a rigid body about an axis that is not passing through its center of mass. It is a mathematical relationship that helps simplify the calculation of the moment of inertia for complex objects.