## Constant Angular Acceleration

Constant angular acceleration describes the relationships among angular velocity, angle of rotation, and time.

### Learning Objectives

Relate angle of rotation, angular velocity, and angular acceleration to their equivalents in linear kinematics

### Key Takeaways

#### Key Points

- The kinematic equations for rotational and/or linear motion given here can be used to solve any rotational or translational kinematics problem in which a and α are constant.
- By using the relationships between velocity and angular velocity, distance and angle of rotation, and acceleration and angular acceleration, rotational kinematic equations can be derived from their linear motion counterparts.
- To derive rotational equations from the linear counterparts, we used the relationships
*a=rα*,*v=rω*, and*x=rθ*.

#### 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.

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:

[latex]\text{v} = \text{v}_0 + \text{at} \\ \text{x} = \text{v}_0 \text{t} + \frac{1}{2} \text{a} \text{t}^2 \\ \text{v}^2 = \text{v}_0^2 + 2\text{ax}[/latex]

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:

[latex]\text{v} = \text{v}_0 + \text{at}[/latex].

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: [latex]\text{a}=\text{r}\alpha [/latex] Where r – radius of curve.Similarly, we have the following relationships between linear and angular values: [latex]\text{v}=\text{r}\omega \\\text{x}=\text{r}\theta [/latex]

By using the relationships *a=rα*, *v=rω*, and *x=rθ*, we derive all the other kinematic equations for rotational motion under constant acceleration:

[latex]\omega = \omega_0 +\alpha \text{t} \\ \theta = \omega_0 \text{t} + \frac{1}{2} \alpha \text{t}^2 \\ \omega^2 = \omega_0^2+2\alpha \theta[/latex]

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.