In this section we will use our new definitions for rotational variables to generate kinematic equations for rotational motion. In addition, we will examine the vector nature of rotational variables and, finally, relate linear and angular variables.
Because our equations defining rotational and translational variables are mathematically equivalent, we can simply substitute our rotational variables into the kinematic equations we have already derived for translational variables. We could go through the formal derivation of these equations, but they would be the same as those derived in One-Dimensional Kinematics. Thus we can simply state the equations, alongside their translational analogues:
v _{f} = v _{o} + at | σ _{f} = σ _{o} + αt | ||
x _{f} = x _{o} + v _{o} t + at ^{2} | μ _{f} = μ _{o} + σ _{o} t + αt ^{2} | ||
v _{f} ^{2} = v _{o} ^{2} + 2ax | σ _{f} ^{2} = σ _{o} ^{2} +2αμ | ||
x = (v _{o} + v _{f})t | μ = (σ _{o} + σ _{f})t |
Now that we have established both equations for our variables and kinematic equations relating them, we can also relate our rotational variables to translational variables. This can sometimes be confusing. It is easy to think that because a particle is engaged in rotational motion, it is not also defined by translational variables. Simply remind yourself that no matter what path a given particle is traveling in, it always has a position, velocity and acceleration. The rotational variables we generated do not substitute for these traditional variables; instead, they simplify calculations involving rotational motion. Thus we can relate our rotational and translational variables.
Recall from our definition of angular displacement that:
Implying that
s = μr |
Thus
v = σr |
Just as linear displacement is equal to angular displacement times the radius, linear velocity is equal to angular velocity times the radius. We can relate α and a , by the same method we used before: differentiating with respect to time.
We must be careful in relating translationa and angular acceleration because only gives us the change in velocity with respect to time in the tangential direction. We know from Dynamics that any particle traveling in a circle experiences a radial force equal to . We must therefore generate two different expressions for the linear acceleration of a particle in rotational motion:
a _{T} | = | αr | |
a _{R} | = | ||
= | σ ^{2} r |
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.
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.
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 :
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.