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# Parallel Axis Theorem, Proof, Definition, Formula, Examples

## Parallel Axis Theorem

According to the parallel axis theorem, a body’s moment of inertia about an axis that passes through its center is equal to the sum of that body’s moment of inertia along its middle axis and the product of that body’s mass times the square of the distance between its two axes.

## Parallel Axis Theorem for Moment of Inertia

There is resistance when we try to adjust a body’s inertia. We call this resistance inertia. The modification may alter the body’s motion in terms of speed or direction. In the absence of external influences, bodies have a tendency to continue traveling in a straight line. Newton’s first law of motion, which is summarised as follows in his book Principia Mathematica, is based on this principle:

Everybody tries to maintain its current state, whether it be one of rest or uniformly moving along in a straight line, using a power of resistance known as the vis insita, or innate force of matter.

## Parallel Axis Theorem Definition

The “Parallel Axis Theorem” states that a body’s moment of inertia about any axis is equal to both its moment of inertia about a parallel axis that passes through its center of mass and the product of that body’s mass and the square root of the distance between the axes.

## Parallel Axis Theorem Formula

The parallel axis theorem formula is as follows: ## Parallel Axis Theorem Proof Derivation

In order to establish the parallel axis theorem, we must demonstrate that a body’s moment of inertia is the product of its moment of inertia about its center, its mass, and the square of the distance between its two axes.

Let’s assume that Ic represents the body’s moment of inertia along an axis that runs through its center. I is the same body’s moment of inertia around the axis that is h away from the center, which is A’B’.

We shall now assume that mass m is present at this position r if r is the distance from this body’s center of gravity.

The distance from A’B’ then becomes h+r. This is the parallel axis theorem’s definitive formula.

When one of the axes travels through the rod’s center and the other, let’s say, runs through one end, the parallel axis theorem can be used to determine the rod’s moment of inertia.

If we use the parallel axis theorem, the equation for a rod’s moment of inertia is given as ML2/12. Observe how:

A rod’s moment of inertia is I = ML2/3.

L is the length of the rod divided by its diameter at one end.

The parallel axis theorem then states: ## Parallel Axis Theorem Equations Important points

The following factors need to be taken into account in order to use the parallel axis theorem.

• Axis A and axis B must be parallel.
• The center of mass of the body must be on axis B.
• They must be as close together as is physically practicable.

## Parallel Axis Theorem Examples

Parallel Axis Theorem Q1: If the moment of inertia of a body along a perpendicular axis passing through its centre of gravity is 50 kg·m2 and the mass of the body is 30 Kg. What is the moment of inertia of that body along another axis which is 50 cm away from the current axis and parallel to it? Use Parallel Axis Theorem Formula

Solution: From the parallel axis theorem,

I = IG + Mb2

I = 50 + ( 30 × 0.52 )

I = 57.5 kg – m2

Parallel Axis Theorem Q2: Calculate the moment of inertia of a rod whose mass is 30 kg and length is 30 cm?

Solution: The parallel axis formula for a rod is given as,

I = (1/12) ML2

plugging in the values we get

I = 0.225 Kg m2.

Parallel Axis Theorem Q3: Calculate the moment of inertia of a stick whose mass is 100 gm and length is 10 cm?

Solution: The parallel axis formula for a rod is given as,

I = (1/12) ML2

plugging in the values we get

I = 0.0000833 Kg m2.

## Parallel Axis Theorem is useful in finding the Area Moment of Inertia

The parallel axis theorem was developed to determine an object’s moment of inertia when the axis passed outside of the central axis. Because of this, calculations are made to be simple, especially for bodies with irregular shapes. It is possible to use the parallel axis theorem with both 2D and 3D objects.

Even while a 3D item cannot be used as a whole, it can be used if it is broken up into smaller pieces, or laminas. The parallel axis theorem is then used to solve each lamina independently. It only works with stiff bodies. Additionally, there is an inertia matrix that changes depending on the person’s point of reference.

### Parallel Axis Theorem- QNAs

Que. What is the statement of the parallel axis?

Ans. According to the parallel axis theorem, a body’s moment of inertia about an axis that is parallel to its axis of mass is equal to the product of its moment of inertia about its axis of mass, the product of mass, and square of the distance between the two axes.

Que. What are the parallel axis and perpendicular axis theorems?

Ans. The parallel axis theorem states that a body’s moment of inertia about any axis is equal to the product of the body’s mass and the square of the perpendicular distance between the two axes, as well as the moment of inertia about a parallel axis that passes through its center of gravity.

Que. What is the parallel axis theorem used in the moment of inertia?

Ans. According to the parallel axis theorem, a body’s moment of inertia about any axis equals its moment of inertia about a parallel axis through its center of mass plus the sum of its mass and the square of the perpendicular distance between its two parallel axes.

Que. Is the parallel axis theorem always true?

Ans. It is really important to note that the parallel axis theorem is only true for Ic or IG – you cannot use the moment of inertia about another point in this formula. It’s also clear that the added term will always be positive (area, mass, and the squared distance cannot be negative).

Que. What is the formula for parallel axis theorem transfer?

Ans. The parallel axis theorem formula is I=Icm+mr2 I = I c m + m r 2

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## FAQs

### What is the statement of the parallel axis?

According to the parallel axis theorem, a body's moment of inertia about an axis that is parallel to its axis of mass is equal to the product of its moment of inertia about its axis of mass, the product of mass, and square of the distance between the two axes.

### What are the parallel axis and perpendicular axis theorems?

The parallel axis theorem states that a body's moment of inertia about any axis is equal to the product of the body's mass and the square of the perpendicular distance between the two axes, as well as the moment of inertia about a parallel axis that passes through its center of gravity.

### What is the parallel axis theorem used in the moment of inertia?

According to the parallel axis theorem, a body's moment of inertia about any axis equals its moment of inertia about a parallel axis through its center of mass plus the sum of its mass and the square of the perpendicular distance between its two parallel axes.

### Is the parallel axis theorem always true?

It is really important to note that the parallel axis theorem is only true for Ic or IG - you cannot use the moment of inertia about another point in this formula. It's also clear that the added term will always be positive (area, mass, and the squared distance cannot be negative).

### What is the formula for parallel axis theorem transfer?

The parallel axis theorem formula is I=Icm+mr2 I = I c m + m r 2