Tru Physics

Chapter 23: Inertia and Angular Momentum

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23.1 Introduction to Rotational Motion

In this chapter, we will explore inertia and angular momentum, two fundamental concepts in rotational motion. These concepts are essential for understanding how objects rotate, conserve angular momentum, and how external forces can change an object’s rotational motion.

A spinning top like this has both a moment of inertia as well as an angular momentum associated with it.
A spinning top like this has both a moment of inertia as well as an angular momentum associated with it.

23.2 Inertia

Inertia is the inherent property of an object that resists changes in its motion. Generally speaking, the greater the mass of an object and the larger its dimensions, the more inertia it possesses.

23.2.1 Rotational Inertia

Rotational inertia, also known as the moment of inertia, is the rotational equivalent of mass in linear motion. Think back to some of the earlier chapters in this course when we discussed Newton’s Second Law. We said that F=ma and that mass is essentially the thing that resists changes in motion when we apply a force to an object. Similarly, an object’s moment of inertia is the thing that resists changes in rotational motion when we apply a torque to an object. We will discuss more on torque in the next chapter. For now, recognize the parallel between mass and torque.

23.2.2 Calculating Rotational Inertia

The moment of inertia (I) of a single point mass (m) located a distance (r) from the axis of rotation is given by the equation:

I = m r^2

For extended objects or systems composed of multiple point masses, the total moment of inertia is the sum of the individual moments of inertia:

I_{sys}= \sum_i m_i r_i^2

Different objects, however, have different methods for calculating the moment of inertia. It is important to recognize that the equations above are for point particles. But what if you had a wooden plank rotating about its length? What if it rotated about its center of mass? Generally, we turn to moment of inertia tables for which formula to use in these different situations.

One great resource for these formulas is Wikipedia.

23.3 Angular Momentum

Angular momentum (\vec{L}) is a vector quantity that describes an object’s rotational motion. It is the rotational equivalent of linear momentum (mass times velocity) and is conserved in the absence of external torques. This is just like the conservation of linear momentum in the absence of external forces.

23.3.1 Calculating Angular Momentum

The angular momentum of a single point mass (m) moving with linear velocity (v) in a circle of radius (r) is given by the equation:

|\vec{L}| = m v r

For extended objects, the angular momentum can be expressed as the product of the moment of inertia (I) and the angular velocity (\omega ):

\vec{L} = I \vec{\omega }

23.4 Conservation of Angular Momentum

The total angular momentum of a closed system is conserved in the absence of external torques. This principle is known as the conservation of angular momentum. It is essential for understanding various phenomena, such as the spinning of ice skaters, the precession of spinning tops, and the orbits of celestial bodies.

Chapter Summary

Inertia and angular momentum are crucial concepts in understanding rotational motion. Inertia describes an object’s resistance to changes in motion, while angular momentum describes its rotational state. Both concepts are essential for predicting and analyzing the behavior of rotating objects and play a central role in various real-world phenomena.

Next, in Chapter 24, we will dive deeper into the concepts of torque and equilibrium, which are closely related to the ideas of inertia and angular momentum.

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Continue to Chapter 24: Torque and Equilibrium
Back to Chapter 22: Momentum, Impulse, and Collisions

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