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### Course: Mechanics (Essentials) - Class 11th > Unit 12

Lesson 2: Motion with constant angular acceleration# Angular kinematics review

Overview of equations and skills for angular kinematics, including how to choose the best angular kinematics formula.

## Equations

$\omega ={\omega}_{0}+\alpha t$ $\theta ={\theta}_{0}+{\omega}_{0}t+{\displaystyle \frac{1}{2}}\alpha {t}^{2}$ ${\omega}^{2}={\omega}_{0}^{2}+2\alpha (\theta -{\theta}_{0})$ $\theta -{\theta}_{0}={\displaystyle \frac{1}{2}}({\omega}_{0}+\omega )t$

Where:

is initial angle${\theta}_{0}$ is final angle$\theta $ is the$t$ is initial angular velocity${\omega}_{0}$ is final angular velocity$\omega $ is angular acceleration$\alpha $

Assumption:

- Angular acceleration
is constant over the time interval.$\alpha $

## Choosing the best rotational kinematic formula

To choose the rotational kinematic formula that's right for your problem, figure out $\omega ={\omega}_{0}+\alpha t$ , to solve for the variables $\omega $ , ${\omega}_{0}$ , $\alpha $ , or $t$ if we knew the values of the other three variables.

**which rotation variable you are not given and not asked to find.**For example, we could use equation 1,Note that each kinematic formula is missing one of the five kinematic variables.

## Common mistakes and misconceptions

**People forget that all the rotational kinematic variables —**${\theta}_{0},\theta ,{\omega}_{o},\omega ,\alpha $ — are vectors and commonly have negative signs.A

*missing negative sign*is a very common source of error. For example, a wheel’s rotation slows down if its angular velocity is counterclockwise (positive direction) and its angular acceleration is clockwise (negative direction). Slowing down is only possible if the angular velocity and acceleration have opposite signs.

## Learn more

For deeper explanations of rotational kinematics, see our video about rotational kinematics.

To check your understanding and work toward mastering these concepts, check out the exercises in this tutorial.

## Want to join the conversation?

- I do not understand that why our speed or acceleration is negative when the object is moving clockwise or virce versa(7 votes)
- Its also due to the fact that the speed(velocity to be precise) and acceleration are vectors and hence the direction in which they are applied matters to us. Since we are working in a rotating system, clockwise being negative and counter-clockwise being positive is analogous to moving forward being positive and moving backward being negative in a translational system(7 votes)

- A fan is rotating at 90rpm. It is then switched off. It stops after 21 revolutions.Calculate the time taken by it to stop assuming that the friction torque is constant.....why is time 2x theta in the solution(1 vote)
- Use equation 4. "w" is zero because the final velocity of the fan is zero. And "theta0" is zero because the initial position can be defined as zero. So it reduces to theta = 1/2 * omega0 * time.(3 votes)

- what if you don't know what the initial angular velocity is? all four formulas listed above seem to include it.(2 votes)
- If you know the values of the other variables in the equation, you shouldn't have too much difficulty in finding the initial angular velocity.

Example:

A wheel speeds up to an angular velocity ofwith an angular acceleration of**32**radians/secondover**4**radians/second squared. What was the initial angular velocity?**3**seconds

In this situation we would use the first angular kinematic formula and solve for the initial angular velocity.

Hope this was helpful!(1 vote)