Angular velocity and the radian

In a nutshell

Angular velocity is the rate of change of angular displacement with respect to time. Linear velocity is the rate of change of linear displacement with respect to time. The number of times an object rotates in a second is given by its frequency and how long it takes to complete one full rotation is given by the time period. Angular velocity is measured in radians per second.

Equations  

Variable definitions

The radian

Consider a circle with a radius of $$r$$. If the arc length is equal to one radius $$r$$, then the angle subtended from the centre of the circle will be equal to $$1 \ radian$$, or $$1 \ rad.$$

In $$180 \degree$$ there is a little over $$3 \ rad$$. In fact there are $$3.14159265359... \ rad$$ which is exactly equal to $$\pi$$.

Note: You need to remember that $$\pi \ rad=180\degree$$ for your exams as you are not given the conversion and it comes up a lot in circular motion! 

To convert from degrees to radians you can use the following equation:

​$$\theta (rad) = \frac {\pi}{180} \times \theta(\degree)$$​​

To convert from radians to degrees you can use this equation:

​

$$\theta (\degree)=\frac {180}{\pi} \times \theta (rad)$$

Example

Convert $$260 \degree$$ into radians.

First, write out the values given and check they are in the correct form:

$$\theta \degree = 260$$

Next, write down the equations needed and rearrange if necessary:

​$$\theta (rad) = \frac {\pi}{180} \times \theta(\degree)$$​

Then, substitute in values into the equation:

​$$\theta (rad) = \frac {\pi}{180} \times 260 \newline \\[0.1in] \theta (rad)=4.5379...$$​​

Make sure to include units and round the lowest number of significant figures of values given by the question:

$$\theta (rad)=4.5 \ rad$$

The angle in radians is $$\underline{4.5 \ rad}$$.

Angular Velocity

Angular velocity $$\omega$$ is defined as the rate of change of  angular displacement $$\theta$$ with time $$t$$:

$$\omega =\frac {\theta}{t}$$​

Time period

The time period $$T$$ is defined as the time taken for one complete revolution of the circle. This is measured in seconds $$s$$.

Frequency

The frequency $$f$$, is defined as how many revolutions are occurring per second and is measured in hertz $$Hz$$.

Time period and frequency are related by the equations:

​$$f= \frac 1T \newline \\[0.1in] T= \frac 1f$$​​

Considering the angular velocity for a complete circle, an object would turn through $$2 \pi \ rad$$ in a time of $$T \ s$$. 

​$$\omega=\frac {2\pi}{T}$$​​

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Substituting in the equation relating frequency and time period, the equation can be also be written in terms of frequency:

​$$\omega=2\pi f$$

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Example

A circular disk is rotating $$45$$ times per second. What is the angular velocity of the disc?

First, write down the values given and check they are in the correct form:

$$f=45 \,Hz$$​​

Next, write down the equations needed and rearrange if necessary:

$$\omega=2\pi f$$​​

Then, substitute the values into the equation:

$$\omega=2\pi \times 45 \newline \\[0.1in] \omega =282.7433...$$​​

Make sure to include units and round the lowest number of significant figures of the values given by the question:

$$angular \ velocity =280 \, rad\, s^{-1}$$​​

The angular velocity is $$\underline{280 \, rad \, s^{-1}}$$.

Linear velocity

Linear velocity is defined as the velocity of an object rotating in uniform circular motion with respect to a linear displacement as opposed to an angular displacement. 

The equation for linear velocity is given as:

​$$v=\frac st$$​​

For a complete revolution, the displacement will be equal to the circumference and the time will be equal to the time period:

​$$v=\frac {2\pi r}{T}$$​​

As $$\omega=\frac {2\pi}{T}$$, the following substitution can be made:

$$v=\omega r$$​​

Example

An object with a velocity of $$14 \, m\,s^{-1}$$ is rotating $$80\, cm$$ about a point. What is the time period of the object?

First, write down the values given and check they are in the correct form:

$$v=14 \,m\,s^{-1} \newline \\[0.1in]r=80\,cm = 0.8\, m$$​​

Next, write down the equations needed:

$$v=\omega r \newline \omega = \frac{2\pi}{T}$$

Substitute in $$\frac{2\pi}{T}$$​ for $$\omega$$ and rearrange for $$T$$:

$$\\[0.1in]v=\frac {2\pi}{T} \times r \newline \\[0.1in]T=\frac {2\pi r}{v}$$​ 

Then, substitute the values into the equation:

$$T=\frac {2\pi \times 0.8}{14} \newline \\[0.1in]T=0.3590...$$​​

Make sure to include units and round the lowest number of significant figures of the values given by the question:

$$Time \ period= 0.36\,s$$​​

The time period is $$\underline{0.36 \, s}$$.