## Key Questions

• In physics you use radians to describe circular motion, in particular you use them to determine angular velocity, $\omega$.
You may be familiar with the concept of linear velocity given by the ratio of displacement over time, as:
$v = \frac{{x}_{f} - {x}_{i}}{t}$
where ${x}_{f}$ is the final position and ${x}_{i}$ is the initial position (along a line).
Now, if you have a circular motion you use the final and initial ANGLES described during the motion to calculate velocity, as:
$\omega = \frac{{\theta}_{f} - {\theta}_{i}}{t}$
Where $\theta$ is the angle in radians.
$\omega$ is angular velocity measured in rad/sec.

(Picture source: http://francesa.phy.cmich.edu/people/andy/physics110/book/chapters/chapter6.htm)

Have a look to other rotational quantities you'll find a lot of ...radians!

• For any $\theta$, the length of the arc is given by the formula (if you work in radians, which you should:

The area of the sector is given by the formula $\frac{\theta {r}^{2}}{2}$

Why is this?
If you remember, the formula for the perimeter of a circle is $2 \pi r$.
In radians, a full circle is $2 \pi$. So if the angle $\theta = 2 \pi$, than the length of the arc (perimeter) = $2 \pi r$. If we now replace $2 \pi$ by $\theta$, we get the formula $S = r \theta$

If you remember, the formula for the area of a circle is $\pi {r}^{2}$.
If the angle $\theta = 2 \pi$, than the length of the sector is equal to the area of a circle = $\pi {r}^{2}$. We've said that $\theta = 2 \pi$, so that means that $\pi = \frac{\theta}{2}$.
If we now replace $\pi$ by $\frac{\theta}{2}$, we get the formula for the area of a sector: $\frac{\theta}{2} {r}^{2}$

• Let's call the cord $A B$ and the centre of the circle $C$

Then if you divide the cord in half at $M$ you get two equal, but mirrored triangles $\Delta C M A$ and $\Delta C M B$. These are both rectangular at $M$. (You should draw this yourself right now !).

$\angle A C M$ is half the central angle that was given
(and $\angle B C M$is the other half)

Then $\sin \angle A C M = \frac{A M}{A C} \to A M = A C \cdot \sin \angle A C M$

Since you know the radius $\left(A C\right)$ and the central angle (remember $\angle A C M =$half of that), you just plug in these values to get an accurate result for half the chord (so don't forget to double it for your final answer)

See examples in explanation

#### Explanation:

Earth's day/night spin about it axis is with

angular speed = $2 \pi$ radian / 24-hour day.

Earths revolution about Sun is owith

angular speed = #2pi) radian / 365.26-day year.

Rotors making electro-mechanical rotations have high angular

speeds of

$k K \pi$ radian / minute, k > 1,

making thousands of rpm ( revolutions / minute ).
,