Precalacus Trig question on linear speed (understand most of it but one part...)

Question

So I'm working on some practice questions and looking at the key our teacher has provided us, but i don't seem to understand one part.

Here's the question:

A wheel's radius is 8 inches rotating at 15 degrees per second. Find the linear speed, angular speed (rpm & radians/second)

So i understand the equation for linear speed is v(velocity)= s(distance)/t(time)

Which would lead me to this -> v=8/15(degrees/second)

but in the key provided, they have it laid out like this:

8in*15(degrees/second)* pi/180

Where does the pi/180 come from?

Answer 1

The linear speed of a rotating wheel is v = ωr where ω is the angular speed in radians per second and r is the radius. Your angular speed is given in degrees per second rather than radians per second. The factor π/180 converts the degrees to radians (there are π radians in 180°).

BTW, your equation v = 8/15 is not right. The wheel does not travel a distance equal to the radius. The correct formula is that the wheel travels a distance equal to its circumference each period, where the period is the amount of time the wheel takes to complete one full revolution.

v = circumference/period

The circumference is 2πr. The period is 2π/ω. Substituting:

V = 2πr/(2π/ω) = ωr