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Just by using our intuition, we can begin to see how rotational quantities like $\theta $ , $\omega $ , and $\alpha $ are related to one another. For example, if a motorcycle wheel has a large angular acceleration for a fairly long time, it ends up spinning rapidly and rotates through many revolutions. In more technical terms, if the wheel’s angular acceleration $\alpha $ is large for a long period of time $t$ , then the final angular velocity $\omega $ and angle of rotation $\theta $ are large. The wheel’s rotational motion is exactly analogous to the fact that the motorcycle’s large translational acceleration produces a large final velocity, and the distance traveled will also be large.
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 $\omega $ , $\alpha $ , and $t$ . To determine this equation, we recall a familiar kinematic equation for translational, or straight-line, motion:
Note that in rotational motion $a={a}_{\text{t}}$ , 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 $\alpha $ is also a constant, because $a=\mathrm{r\alpha}$ . Now, let us substitute $v=\mathrm{r\omega}$ and $a=\mathrm{r\alpha}$ into the linear equation above:
The radius $r$ cancels in the equation, yielding
where ${\omega}_{0}$ is the initial angular velocity. This last equation is a kinematic relationship among $\omega $ , $\alpha $ , 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.
Kinematics for rotational motion is completely analogous to translational kinematics, first presented in One-Dimensional Kinematics . Kinematics is concerned with the description of motion without regard to force or mass. We will find that translational kinematic quantities, such as displacement, velocity, and acceleration have direct analogs in rotational motion.
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 | |
---|---|---|
$$\theta =\overline{\omega}t$$ | $$x=\stackrel{-}{v}t$$ | |
$$\omega ={\omega}_{0}+\mathrm{\alpha t}$$ | $$v={v}_{0}+\text{at}$$ | (constant $\alpha $ , $a$ ) |
$$\theta ={\omega}_{0}t+\frac{1}{2}{\mathrm{\alpha t}}^{2}$$ | $$x={v}_{0}t+\frac{1}{2}{\text{at}}^{2}$$ | (constant $\alpha $ , $a$ ) |
$${\omega}^{2}={{\omega}_{0}}^{2}+2\text{\alpha \theta}$$ | $${v}^{2}={{v}_{0}}^{2}+2\text{ax}$$ | (constant $\alpha $ , $a$ ) |
In these equations, the subscript 0 denotes initial values ( _{ ${\theta}_{0}$ } , ${x}_{0}$ , and ${t}_{0}$ are initial values), and the average angular velocity $\stackrel{-}{\omega}$ and average velocity $\stackrel{-}{v}$ are defined as follows:
The equations given above in [link] can be used to solve any rotational or translational kinematics problem in which $a$ and $\alpha $ are constant.
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