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# Angles in Radians and Angular Speed versus Linear Speed

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## Angles in Radians and Angular Speed versus Linear Speed

We usually measure angles in degrees.

**360° = 1 rotation**

But it's not the most convenient way to measure angles in circular motion.

Here's an alternative: **Radians.** The radius of a circle and its circumference are related by the equation...

Circumference = 2πr

So the factor that allows you to convert from circumference (distance travelled around the arc of the circle) to radius is 2π.

So in this way, 2π describes a whole circle, just as 360° describes a whole circle.

360° ≡ 2π radians and

180° ≡ π radians

**In fact, as long as you use angles in radians you can write this general equation:**

s = rθ

**Where:**

s = arc length covered

r = radius of the circle

θ = angle in radians

**Example:**

**Convert the following angles from degrees into radians.**

90°, 135°, 330°

**Answer:**

180° ≡ π radians.

So multiply any angle in degrees byto find the same angle in radians.

**Example:**

**What angle in degrees has a car travelled around a circular track if the track has a radius of 100 m and the distance covered by the car is 470 m?**

**Answer:**

**Convert to degrees:**

* Note:* We had to turn the conversion factor upside down to convert from radians to degrees.

**Question:**

In linear or straight-line motion, we measure speed by looking at how much **distance** is covered each second. You can do that in circular motion too, but it's often better to use **angular speed**, ω.

Angular speed measures the angle of a complete circle (measured in radians) covered per second.

For instance,

**Where:**

θ = angle of rotation in radians

t = time taken in seconds.

If you consider that the time taken for a complete rotation is the period, T, then

because 2π is the angle covered (in radians) when you do a complete circle.

Remembering thatyou can also write this as

ω = 2πf

**Example:**

**An old record player spins records at 45rpm (revolutions per minute). For a point on the circumference (radius = 10cm) calculate the angular speed in rad s ^{-1}.**

**Answer:**

45rpm = 45/60 = 0.75 revolutions per second = f

Angular speed = ω = 2πf x 0.75 = 4.7 rad s^{-1}

**Question:**

**The wheel of a car rotates at 10 revolutions per second as the car travels along. The radius of the rubber on the tyre is 20cm.**

If you are going round in a circle of radius, r, and you are travelling at a linear speed, v ms^{-1}:

The distance covered in 1 rotation = 2πr

The time for one rotation = T, the period.

These equations allow you to relate **angular and linear speed.**

**Question:**

**A shot put is swung round at 1 revolution per second. The athlete's arm is 60cm long.**