Video Transcript
The wheels of a moving car rotate 13.5 times per second. What is the angular velocity of a point on the wheel that is not at the axis of the rotation of the wheel?
Let’s start by drawing a diagram. Here’s a wheel of a moving car. We should recall that, except for the very center axis of rotation, we could choose any point on the wheel and find that it rotates at the same angular velocity as any other point. So let’s just pick somewhere, say this point here. Note that we would have to be more specific if we were considering linear velocity, which does depend upon how far a point is from the axis of rotation. But since we’re talking about angular velocity, we don’t need to stress over specifying exactly where our point of interest is.
Moving on, recall that angular velocity, represented by 𝜔, is defined as a change in angular displacement, 𝛥𝜃, divided by some change in time, 𝛥𝑡. One more thing that’s important to remember is that angular velocity should be expressed in radians per second. Presently, we know that this wheel rotates 13 and a half times every second or that it makes 13.5 revolutions per second. To express this as a proper angular velocity, we’ll need to convert revolutions to radians.
Now, we know that one revolution means one full turn around a circle, which measures as two 𝜋 radians. Now, we can use this equality to write this conversion factor. Because its numerator and denominator are equivalent, the whole factor itself is just equal to one. So we can multiply it by 13.5 revolutions without actually changing its value.
No matter what, this wheel is still rotating 13 and a half times every second. We just want to express these rotations in radians, which is why we write this conversion factor with two 𝜋 radians in the numerator and revolutions in the denominator. This way, revolutions cancel out of this expression completely. And we know that the change in angular displacement is 13.5 times two 𝜋 radians. 13.5 times two equals 27. So let’s write this as 27 times 𝜋 radians. This is 𝛥𝜃, and let’s not forget about 𝛥𝑡. We know the wheel turns through 27 times 𝜋 radians per second. So 𝜔 equals 27 times 𝜋 radians per second.
Evaluating this further, let’s multiply 27 by 𝜋, and we have 84.823 and so on radians per second. Rounding to the nearest whole number gives 85 radians per second, and we have our answer. This is the angular velocity of a point on the wheel.
