Chapter 5: Rotational motion

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.

A wheel with ten spokes is shown. A point on the wheel at the end of one spoke is labeled with a capital P. An arrow extends from this point, perpendicular to the spoke. An arced arrow below the wheel indicates that the wheel is rotating.
An object traveling in a circular path has a velocity that is directed tangent to the circle.

Example

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

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.

A circle with two dashed lines extending from the center. Two dots on one dashed line represent the starting positions of people standing on a rotating circle, and dots on the other dashed line represent their ending positions. Arcs connect the starting position to the ending position for each dot.
A person on the outside of a merry go round has a higher tangential speed than someone closer to the center, because they travel a larger distance in the same amount of time.

practice

practice

Practice

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.

This image shows a wheel that rolls without slipping. The wheel is represented by a circle and the ground is represented by a horizontal line. An arrow extends horizontally from the center of the circle, pointing to the right. Farther to the right is a circle made of a dashed line, representing the wheel after it has made one complete rotation. The distance between the centers of the circles is indicated as being equal to the circumference of the circle.
A wheel that rolls without slipping

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.

Practice