MHB Finding revolution/minute of a wheel traveling in a speed of 15 miles/hour

[bsmithysmith]

A truck with 32- inch- diameter wheels is traveling in a speed of 15 miles/hour. Find the angular speed of the wheels in radians/minute. How many revolutions per minute does the wheel make?

My teacher mentions that we are trying to find the angular speed in this. Another question is how do I distinguish using angular or linear speed/formula for this equation? I know that angular speed = theta/time and linear speed = arclength/time, but when he explained the question, he made up a newer looking equation: v = (2pi(r)w)/theta. I don't understand this at all and how to get the answer.

 

[MarkFL]

Let's begin with the equation for linear distance, speed and time:

$$d=vt$$

When the truck's wheel moves through an angle $\theta$, then the truck will move a linear distance equal to:

$$d=r\theta$$

This comes from the arc-length of a circular sector.

And so we have:

$$r\theta=vt$$

Now, the angular velocity $\omega$ is defined as the change of the angle $\theta$ with respect to time, and for constant speeds, this is:

$$\omega\equiv\frac{\theta}{t}$$

Now, recall we have:

$$r\theta=vt$$

Dividing through by $t$, we obtain:

$$v=r\frac{\theta}{t}=r\omega$$

or:

$$\omega=\frac{v}{r}$$

You are given $v$ and $r$ and so you can find $\omega$ from this formula, but be mindful of your units, because $v$ is given in miles per hour, but you want time in minutes. Can you proceed?

 

[bsmithysmith]

Sorry it took awhile to reply but yes! That really helped a lot! Thank you so much for that! I asked my friend who's taking a physics class and he mentioned that the linear speed formula is similar to something he learned, except it's not:

$$v=(radius(theta))/time$$

but

$$v=distance/radius$$

I'm still very edgy and somewhat having trouble with trigonometry, my teacher's back is always blocking what he's doing.