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The Power of Rotational Equations

With these equations we can describe the motion of any given particle through rotational and translational variables. So why even bother with rotational variables if everything can be expressed in terms of the more familiar linear variables? The answer lies in the fact that every particle in a rigid body has the same value for rotational variables. This characteristic makes rotational variables a far more powerful means of predicting the motion of rotating bodies, and not just particles.

Vector Notation of Rotational Variables

Every equation we have derived so far has been in terms of the magnitude of our rotational variables. But what about their direction? Can we give our variables both magnitude and direction? It would seem as though the direction of our rotational variables would be the same as our linear ones. For instance, it would make sense to make the direction of angular velocity always tangent to the circle through which the particle travels. However, with this definition, the direction of σ is always changing, even if the particle is moving with constant angular velocity. Clearly, such inconsistency is a problem; we must define the direction for our variables in a new way.

For reasons too complicated to discuss here, angular displacement μ cannot be represented as a vector. However, σ and α can, and we shall describe how to find their direction through the right hand rule.

Right Hand Rule

Take your right hand, curl your fingers, and stick your thumb straight up. If you let the curl of your fingers follow the path of the rotating particle or body, your thumb will point in the direction of the angular velocity of the body. This way, the direction is constant throughout the rotation. Below are shown a few examples of rotation, and of the resultant direction of σ:

Figure %: Three different directions of rotation, shown with direction of σ

Angular acceleration is defined in a similar way. If the magnitude of the angular velocity increases, then the angular acceleration is in the same direction as the angular velocity. Conversely, if the magnitude of the velocity decreases, the angular acceleration points in the direction opposite the angular velocity.

Though the direction of these vectors may seem trivial for now, they become quite important when studying concepts such as torque and angular momentum. Now, equipped with kinematic equations for rotational motion, relations between angular and linear variables, and a sense of the vector notation of rotational variables, we are able to develop and explore the dynamics and energetics of rotational motion.

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