Simply by using our intuition, we can begin to see the interrelatedness of rotational quantities like* θ* (angle of rotation), *ω*(angular velocity) and *α* (angular acceleration). For example, if a motorcycle wheel has a large angular acceleration for a fairly long time, it ends up spinning rapidly and rotating through many revolutions. The wheel’s rotational motion is analogous to the fact that the motorcycle’s large translational acceleration produces a large final velocity, and the distance traveled will also be large.

### Kinematic Equations

Kinematics is the description of motion. We have already studied kinematic equations governing linear motion under constant acceleration:

\[\begin{align} \mathrm{v} & \mathrm{=v_0+at} \\ \mathrm{x} & \mathrm{=v_0t+\frac{1}{2}at^2} \\ \mathrm{v^2} & \mathrm{=v_0^2+2ax} \end{align}\]

Similarly, 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 use the corresponding equation for linear motion:

\[\mathrm{v=v_0+at.}\]

As in linear kinematics where we assumed *a* is constant, here we assume that angular acceleration *α* is a constant, and can use the relation: \(\mathrm{a=rα}\) Where r – radius of curve.Similarly, we have the following relationships between linear and angular values:

\[\begin{align} \mathrm{v} & \mathrm{=rω} \\ \mathrm{x} & \mathrm{=rθ} \end{align}\]

By using the relationships \(\mathrm{a=rα, v=rω,}\) and \(\mathrm{x=rθ}\), we derive all the other kinematic equations for rotational motion under constant acceleration:

\[\begin{align} \mathrm{ω} & \mathrm{=ω_0+αt} \\ \mathrm{θ} & \mathrm{=ω_0t+\frac{1}{2}αt^2} \\ \mathrm{ω^2} & \mathrm{= ω_0^2+2αθ} \end{align}\]

The equations given above can be used to solve any rotational or translational kinematics problem in which a and α are constant. shows the relationship between some of the quantities discussed in this atom.

**Linear and Angular**: This figure shows uniform circular motion and some of its defined quantities.

## Key Terms

**kinematics**: The branch of mechanics concerned with objects in motion, but not with the forces involved.
**angular**: Relating to an angle or angles; having an angle or angles; forming an angle or corner; sharp-cornered; pointed; as in, an angular figure.

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