69 Kinematics of Rotational Motion
Learning Objectives
By the end of this section, you will be able to:
- Observe the kinematics of rotational motion.
- Derive rotational kinematic equations.
- Evaluate problem solving strategies for rotational kinematics.
Kinematics is the description of motion. The kinematics of rotational motion describes the relationships among rotation angle, angular velocity, angular acceleration, and time. Let us start by finding an equation relating ω, α, and t. To determine this equation, we recall a familiar kinematic equation for translational, or straight-line, motion:
[latex]v={v}_{0}+{at}\\[/latex] (constant a)
Note that in rotational motion a = at, and we shall use the symbol a for tangential or linear acceleration from now on. As in linear kinematics, we assume a is constant, which means that angular acceleration α is also a constant, because a = rα. Now, let us substitute v = rω and a = rα into the linear equation above:
rω = rω0 + rat.
The radius r cancels in the equation, yielding
ω = ω0 + at. (constant a)
where ω0 is the initial angular velocity. This last equation is a kinematic relationship among ω, α, and t —that is, it describes their relationship without reference to forces or masses that may affect rotation. It is also precisely analogous in form to its translational counterpart.
Making Connections
Starting with the four kinematic equations we developed in One-Dimensional Kinematics, we can derive the following four rotational kinematic equations (presented together with their translational counterparts):
| Rotational | Translational | |
|---|---|---|
| [latex]\theta =\bar{\omega }t\\[/latex] | [latex]x=\bar{v}t\\[/latex] | |
| ω = ω0 + αt | v = vo + at | (constant α, a) |
| [latex]\theta ={\omega }_{0}t+\frac{1}{2}{\alpha t}^{2}\\[/latex] | [latex]x={v}_{0}t+\frac{1}{2}{\text{at}}^{2}\\[/latex] | (constant α, a) |
| ω2 = ω02+ 2αθ | v2 = vo2 + 2ax | (constant α, a) |
In these equations, the subscript 0 denotes initial values (θ0, x0, and t0 are initial values), and the average angular velocity [latex]\bar{\omega}\\[/latex] and average velocity [latex]\bar{v}\\[/latex] are defined as follows:
[latex]\bar{\omega}=\frac{{\omega }_{0}+\omega }{2}\text{ and }\overline{v}=\frac{{v}_{0}+v}{2}\\[/latex].
The equations given above in Table 1 can be used to solve any rotational or translational kinematics problem in which a and α are constant.
Problem-Solving Strategy for Rotational Kinematics
- Examine the situation to determine that rotational kinematics (rotational motion) is involved. Rotation must be involved, but without the need to consider forces or masses that affect the motion.
- Identify exactly what needs to be determined in the problem (identify the unknowns). A sketch of the situation is useful.
- Make a list of what is given or can be inferred from the problem as stated (identify the knowns).
- Solve the appropriate equation or equations for the quantity to be determined (the unknown). It can be use