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# Moment of Inertia

The moment of inertia or mass moment of inertia is a scalar quantity that measures a rotating body’s resistance to rotation. The higher the moment of inertia, the more resistant a body is to angular rotation. A body is usually made from several small particles forming the entire mass. The mass moment of inertia depends on the distribution of each individual…

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# Moment of Inertia

Moment of Inertia
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Nie wieder prokastinieren mit unseren Lernerinnerungen. The moment of inertia or mass moment of inertia is a scalar quantity that measures a rotating body’s resistance to rotation. The higher the moment of inertia, the more resistant a body is to angular rotation. A body is usually made from several small particles forming the entire mass. The mass moment of inertia depends on the distribution of each individual mass concerning the perpendicular distance to the axis of rotation. However, in physics, we usually assume that the mass of an object is concentrated at a single point called the centre of mass.

## Moment of inertia equation

Mathematically, the moment of inertia can be expressed in terms of its individual masses as the sum of the product of each individual mass and the squared perpendicular distance to the axis of rotation. You can see this in the equation below. I is the moment of inertia measured in kilogram square metres (kg·m2), m is the mass measured in kilograms (kg), and r is the perpendicular distance to the axis of rotation measured in metres (m).

$I = \sum_i^n m \cdot r^2_i$

We can also use the equation below for an object whose mass is assumed to be concentrated to a single point. The image shows the distance of the axis of rotation r. Fig. 1 - Diagram showing the distance of the axis of rotation r

$I = m \cdot r^2$

## Where did the moment of inertia come from?

Newton’s law states that the linear acceleration of an object is linearly proportional to the net force acting upon it when mass is constant. We can state this with the equation below, where Ft is the net force, m is the object’s mass, and at is the translational acceleration.

$F_t = m \cdot a_t$

Similarly, we use torque for rotational motion, which is equal to the product of the rotational force and the perpendicular distance to the axis of rotation. However, the translational acceleration for rotational motion is equal to the product of angular acceleration α and radius r.

$\alpha_t = r \cdot \alpha \frac{T}{r} = m \cdot r \cdot \alpha \Rightarrow T = m \cdot r^2 \cdot \alpha$

The moment of inertia is the reciprocal of the mass in Newton’s second law for linear acceleration, but it is applied to angular acceleration. Newton’s second law describes the torque acting on a body, which is linearly proportional to the mass moment of inertia of a body and its angular acceleration. As seen in the derivation above, the torque T is equal to the product of the moment of inertia I and angular acceleration $$\alpha$$.

$T = I \cdot \alpha$

## Moments of inertia for different shapes

The moment of inertia is different for and specific to each object’s shape and axis. Due to the variation in geometric shapes, a moment of inertia is given for various commonly used shapes, which you can see in the image below. Fig. 2 - Moment of inertia for different shapes

We can calculate the moment of inertia for any shape by integration (about the x-axis) of the product of the equation, which describes the width or thickness d, the rate of change of y, and A multiplied by the squared distance to the axis.

$I = \int dA \cdot y^2$

The greater the thickness, the greater the moment of inertia.

## Examples of calculating the moment of inertia

A thin disk with a 0.3 m diameter and a total moment of inertia of 0.45 kg·m2 is rotating about its centre of mass. There are three rocks with masses of 0.2 kg on the outer part of the disk. Find the total moment of inertia of the system.

Solution

The radius of the disk is 0.15 m. We can calculate the moment of inertia of each rock as

$I_{rock} = m \cdot r^2 = 0.2 kg \cdot 0.15 m^2 = 4.5 \cdot 10^{-3} kg \cdot m^2$

Hence, the total moment of inertia will be

$I_{rocks} + I_{disk} = (3 \cdot I_{rock})+I_{disk} = (3 \cdot 4.5 \cdot 10^{-3} kg \cdot m^2) + 0.45 kg \cdot m^2 = 0.4635 kg \cdot m^2$

An athlete is sitting in a rotating chair holding a training weight of 10kg in each hand. When will the athlete be more likely to rotate: when he extends his arms far from his body or when he retracts his arms close to his body?

Solution

When the athlete extends his arms, the moment of inertia increases as the distance between the weight and his axis of rotation increases. When the athlete retracts his arms, the distance between the weights and the axis of rotation decreases, and so does the moment of inertia.

Therefore, the athlete is more likely to rotate when he retracts his hands as the moment of inertia will be smaller and the body will have less resistance to rotating.

A very thin disk with a diameter of 5cm is rotating about its centre of mass, and another thicker disc with a diameter of 2 cm is rotating about its centre of mass. Which of the two disks has a larger moment of inertia?

Solution

The disc with the larger diameter will have a larger moment of inertia. As the formula suggests, the moment of inertia is proportional to the squared distance to the axis of rotation, hence the greater the radius, the larger the moment of inertia.

## Moment of Inertia - Key takeaways

• The moment of inertia is a measure of a rotating object’s resistance to rotation. It is dependent on mass and the distribution of its mass about its axis of rotation.

• The moment of inertia is the reciprocal of mass in Newton’s second law applied for rotation.

• The moment of inertia is different and specific to each object’s shape and axis.

Images

Rotational inertia. https://web2.ph.utexas.edu/~coker2/index.files/RI.htm

The moment of inertia can be calculated by the sum of the product of individual masses of an object and their respective squared perpendicular distance to the axis of rotation.

The moment of inertia or mass moment of inertia is a scalar quantity that measures a rotating body’s resistance to rotation. The higher the moment of inertia, the more difficult it is for a body to rotate and vice versa.

The moment of inertia is the reciprocal of the mass in Newton’s second law for linear acceleration, but it is applied for angular acceleration.

## Moment of Inertia Quiz - Teste dein Wissen

Question

What is the moment of inertia?

The moment of inertia is a measure of an object’s difficulty to rotate about its axis of rotation.

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Question

What is the definition of the moment of inertia?

The moment of inertia is the product of an object’s mass and the distance of its distributed mass to its axis of rotation.

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Question

What is the equation we use to calculate the moment of inertia?

I=m·r2

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Question

What does r represent in the equation of moment of inertia?

The distance of the distributed mass of an object to its axis of rotation.

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Question

What is the unit of moment of inertia?

The unit of moment of inertia is kg·m2.

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Question

How does the moment of inertia relate to Newton’s second law?

The moment of inertia is the reciprocal of the mass in Newton’s second law.

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Question

What is the relationship between Newton’s second law and Newton’s law of rotation?

F=m·a and T=I·a. They have the same form, hence, the moment of inertia is the reciprocal of mass in Newton’s second law for linear acceleration.

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Question

Does a higher moment of inertia mean that a body is more likely to rotate?

No, a higher moment of inertia does not mean that a body is more likely to rotate.

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Does a high moment of inertia mean that a body is less likely to rotate?

Yes.

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An ice skater is trying to perform aerobics while ice skating. How can he ensure that he remains stable once he performs his rotations?

He can extend his arms and legs as far away from the centre of mass as possible.

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When someone is rotating and suddenly extends his arms away from his body, how is the moment of inertia affected?

The moment of inertia is increased as the distance between the distribution of mass and the axis of rotation is increased.

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A training weight has two disks connected with a rod. When will the training weight be easier to rotate?

When the two weight discs are placed in the middle of the rod.

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Two people are sitting in a small boat How should they sit in order to resist rotation of the boat?

They can sit on the two ends of the boat.

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Question

An object has a moment of inertia of 60kg·m2. How will the moment of inertia be affected if the perpendicular distance to the axis of rotation or its distributed mass is doubled?

The total moment of inertia will be quadrupled.

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