Applications of Radian Measure

Key Questions

What are some applications of using radian measure?

In physics you use radians to describe circular motion, in particular you use them to determine angular velocity, $ω$ $omega$ .
You may be familiar with the concept of linear velocity given by the ratio of displacement over time, as:
$v = \frac{x_{f} - x_{i}}{t}$ $v=(x_f-x_i)/t$
where $x_{f}$ $x_f$ is the final position and $x_{i}$ $x_i$ 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:
$ω = \frac{θ_{f} - θ_{i}}{t}$ $omega=(theta_f-theta_i)/t$
Where $θ$ $theta$ is the angle in radians.
$ω$ $omega$ 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!

Gió · 1 · Jan 11 2015
How do you calculate the length of an arc and the area of a sector?

For any $θ$ $theta$ , the length of the arc is given by the formula (if you work in radians, which you should:
![mathwarehouse.com]()
The area of the sector is given by the formula $\frac{θ r^{2}}{2}$ $(theta r^2)/2$

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

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

Nico Ekkart · 5 · Dec 19 2014
How do you approximate the length of a chord given the central angle and radius?

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

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

$angle ACM$ is half the central angle that was given
(and $angleBCM$ is the other half)

Then $sin angle ACM=(AM)/(AC) ->AM=AC*sin angle ACM$

Since you know the radius $(AC)$ and the central angle (remember $angleACM=$ 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)

MeneerNask · 1 · Jan 13 2015
What are some example problems involving angular speed?

Answer:

See examples in explanation

Explanation:

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

angular speed = $2pi$ 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

$kKpi$ radian / minute, k > 1,

making thousands of rpm ( revolutions / minute ).
,

A. S. Adikesavan · 1 · Aug 10 2018

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Applications of Radian Measure

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