5.2 Connecting translation & rotation
5.2.1 Tangential speed
Consider an object traveling at some speed \(v\) in a circular path, such as a ball being swung around on a string. At any given instant, an object moving along a circular path has some velocity pointing in a direction tangential to the path (see the figure below). This is called the tangential velocity, \(\vec{v}_t\). Since the object is traveling in a circle, the direction is constantly changing. However, the magnitude of the tangential velocity—called the tangential speed—is given by:
\[
v_t = r\omega \tag{5.5}
\]
This has units of meters per second. The tangential speed is what an instrument such as a baseball pitching coach’s radar gun would tell you, if used to measure the motion of an object moving in a circular path.
Example 5.2
The radius of the Earth at the equator is 6378 km. Given that Earth makes one rotation each day, find the tangential speed of a person at the equator.
Use the relationship between tangential speed, angular speed, and radius:
\[
\begin{align*}
v_t &= r\omega \\
&= \left(6378\ \textrm{km}\right)\left(\frac{2\pi\ \textrm{rad}}{24\ \textrm{hr}}\right) \\
&= 1670\ \textrm{km/hr}
\end{align*}
\]
That’s about a thousand miles per hour!
Example 5.3
You and your friend are riding on a merry-go-round. If your friend is twice as far from the center of the merry-go-round as you are, how many times faster are they traveling?
Since you are both on the same rotating object, your angular speed is the same. In the relationship \(v_t = r\omega\), we see that if you double \(r\) and keep \(\omega\) the same, you double \(v_t\). (When two variables are related like this they are said to be directly proportional.) So, your friend is traveling twice as fast as you are.
Notice that a point on farther from the axis of rotation is actually traveling faster than a point closer in. Think about it this way: in the time it takes to make one complete rotation, your friend (who is closer to the outer edge of the merry-go-round) travels a much larger distance than you. The circumference of your friend’s circular path is larger than that of yours. Since your friend traveled a larger distance in the same amount of time, they are traveling at a higher speed. The figure below shows how the length of the paths around a circle differ depending on distance from the center.
practice 5.3
practice 5.4
Practice 5.5
5.2.2 Rolling motion
We have been talking about rotation around a fixed axis. But what if the axis of rotation was moving forward, like a bicycle wheel as you ride in a straight line? The motion of the wheel is a combination of rotational and translational motion.
The figure above shows a wheel with a radius \(R\). If the wheel rolls without slipping, as it makes one complete revolution it moves forward by a distance equal to the circumference. It does this in some time \(T\). The speed of the axle is then
\[
v = \frac{2\pi R}{T}
\]
Since it takes time \(T\) to make one complete rotation of \(2\pi\) radians, \(\frac{2\pi}{T}\) is the angular speed of the wheel. This gives us
\[
v = R\omega
\]
which is the tangential speed of a point on the rim. The axle is moving forward with a translational speed equal to the tangential speed of a point on the rim. This is true for any object that is rolling without slipping.