We are all used to measuring angles in degrees and we know their are 360° in a circle.

			Did you ever wonder why we divide the circle into 360 parts?  It comes from the fact that there are 365 days in a year.  So, one degree is about how much the earth moves daily in its orbit around the sun.

			At some point it was decided that we should measure angles in some unit that did not depend specifically on the earth or the sun, but just depended on the properties of a circle.

			You learned in trigonometry that angles can also be measured in radians by drawing the angle from the center of a circle to the edge of the circle and dividing the length of the arc formed by the sides of the angle by the radius of the circle:  Angle in radians = (arc length)/radius or θ = S/R.  If the arc length S equals the radius R, then θ = R/R = 1 radian.  If the arc length is the whole circumference of the circle, then θ = (2πR)/R = 2π radians, so there are 2π radians in a circle.  Hence 2π radians = 360° or π radians = 180°.

			We can convert an angle from degrees to radians by multiplying it by π/180°.

			We can convert an angle from radians to degrees by multiplying it by 180°/π.

			For many purposes it's simpler to work in radians than in degrees.

			For example, if I know a laser beam is spreading out with an angle of 1 milliradian = 0.001 radian, I can estimate the width of the beam at any distance simply by multiplying the angle by the distance.  In other words, S = Rθ.  So, at a distance of 100 m, the length of a soccer field, the beam would have a width of S = (100m)(0.001) = 0.1 m.  At a distance of a mile, the beam would have a width of S = (1 mile)(0.001) = 0.001 mile ≈ 5 feet.

			An object moving in a circular path has an angular position which is constantly changing in time.  The rate of change of the angular position with time is related by the circle's radius to the speed with which the object is moving on the edge of the circle.  In other words, if S = Rθ, then ∆S/∆t = R(∆θ/∆t) or v = Rω.  The rate at which the object moves along the arc is the tangential velocity v (usually measured in meters/second), and the rate at which the angle changes is called angular velocity (usually measured in radians/second) and is represented by the Greek letter omega ω.

			Using this formula we can easily find how fast (in meters/second) the earth is moving in its orbit around the sun.  The angular velocity of the earth around the sun is (2π radians)/(seconds in a year) = (2π)/(3.16x107) ≈ 2 x 10-7 radians/second.  The distance R of the earth from the sun is 1.496 x 1011 m, so v = Rω = (1.496 x 1011m)(2 x 10-7radians/second) ≈ 3 x 104 m/s = 30,000 m/s ≈ 60,000 miles/hour.

			Using the same formula, we can find the angular velocity ω if we know v and R.  If we know a car is going around a curve of radius R = 100 m at a speed of 20 m/s, then its angular velocity is ω = v/R = (20m/s)/(100m) = 0.20 radians/s.

			If an object is changing its speed as it goes around a circle, then it has angular acceleration.  Since a = ∆v/∆t we can write a = ∆v/∆t = R(∆ω/∆t) = Rα, where the rate of change of angular velocity ∆ω/∆t is denoted by the Greek letter alpha α.

			Suppose an object has an angular acceleration of α = 0.2 radians/s2 as it goes around a circle of radius R = 10m.  What is its tangential acceleration in m/s2?  a = Rα = (10m)(0.2radians/s2) = 2 m/s2, meaning it changes its speed by 2m/s every second.

			Suppose a pilot accelerates at 5 m/s2 as he flies around a circular arc of radius 1000 m?  What is his angular acceleration?  α = a/R = (5m/s2)/(1000m) = 0.005 radians/s2.

			The slope of the displacement-time graph is the linear velocity.

			The slope of the velocity-time graph is the linear acceleration.

			The area under the linear acceleration-time graph is the change in velocity.

			The area under the velocity-time graph is the change in displacement in meters.

			The slope of the angle-in-radians-time graph is the angular velocity.

			The slope of the angular-velocity-time graph is the angular acceleration.

			The area under the angular-acceleration-time graph is the change in angular velocity.

			The area under the angular-velocity-time graph is the change in angle in radians.

			a = force/mass = Fnet/m

			α = torque/(moment of inertia) = τ/I