Uniform Circular Motion and its Important Quantities

• The movement of the moon around the earth,

• a circulating electric charge in the magnetic field

• or a looping.

These are all nearly uniform circular motions. Let's get to know this simplest form of circular motion a little better.

Consider a body that has mass \(m\) and is moving in a circle. We say: The body moves on a circular path. The circular path has the radius \(r\). This is the distance of the body from the center of the circular path.

Let's look at a snapshot at a particular time \(t_1\). The body is then located, for example, at the point \(S_1\) on the circular path. At a later time \(t_2\) the body is at \(S_2\). The time it took the body to get from \(S_1\) to \(S_2\) is the time span \( \Delta t ~=~ t_2 - t_1 \).

The body has traveled a distance \(\Delta s\), called arc length, as it moved from \(S_1\) to \(S_2\). The average speed \(\bar{v}\) is distance traveled \(\Delta s\) per time \(\Delta t\):

If we choose \(\Delta t\) to be infinitely small (we note an infinitely small time span as \(\text{d}t\)), then the distance traveled \(\Delta s\) also becomes infinitely small: \(\text{d}s\)). The average velocity \(\bar{v}\) is then also the instantaneous velocity \(v\):

The instantaneous velocity \(v\) is the derivative of the position \(s\) with respect to the time \(t\). Eq. 2 is only the magnitude of the velocity. We can assign a direction to the magnitude and thus transform the velocity \(v\) into a vector quantity \(\boldsymbol{v}\). We represent vectors in bold to distinguish the vector \(\boldsymbol{v}\) from its magnitude \(v\).

The velocity vector touches the circular path at the point \(S_1\). We say: The velocity vector \(\boldsymbol{v}\) is tangential to the circular path. We also call this tangential instantaneous velocity trajectory velocity. We can therefore assign an orbital velocity to the body at each point of the circular path. At the point \(S_1\) the body has the orbital velocity \(\boldsymbol{v}_1\) and at the location \(S_2\) the orbital velocity is \(\boldsymbol{v}_2\).

A uniform circular motion is characterized by the fact that the magnitude \(v\) of the orbital velocity is constant at every point:

The direction of the velocity is different at each point of the circular path. At the point \(S_1\) the orbital velocity \(\boldsymbol{v}_1\) points in a different direction than the orbital velocity \(\boldsymbol{v}_2\) at the point \(S_2\). Also, if the magnitude of the orbital velocity does not change as the body moves, the direction (due to the curved path) and thus the velocity vector \(\boldsymbol{v}_1\) changes to \(\boldsymbol{v}_2\).

The change \(\delta \boldsymbol{v} \) of the velocity vector is given by:

The average acceleration \( \boldsymbol{a} \) is defined as the change \(\delta \boldsymbol{v}\) per time span \(\delta t\):

Here in Eq. 5, if we consider an infinitesimally small time span \(\text{d} t\), the velocity change \(\text{d} \boldsymbol{v}\) also becomes infinitesimally small. The average acceleration becomes the instantaneous acceleration \(\boldsymbol{a}\) (exact acceleration at a point of the circular path). It corresponds to the derivative of the orbital velocity with respect to time:

Since the velocity changes during a circular motion, the body experiences an acceleration according to Eq. 6.

Here, we do not consider an infinitely small time span, but a finitely small time span \(\Delta t\). So we are dealing here with the average velocity 1. However, we choose the time span so small that the body covers only a small distance \( \Delta s \). Then \( \boldsymbol{v}_1 \) and \( \boldsymbol{v}_2\) are approximately parallel to each other. Thus \( \delta \boldsymbol{v} \) is perpendicular to the two velocity vectors.

If \(\Delta \boldsymbol{v}\) is perpendicular to \(\boldsymbol{v}_1\), then \(\Delta \boldsymbol{v}\) points to the center of the circle. \(\delta \boldsymbol{v}\) is parallel to the radius \(r\).

According to Eq. 5, \(\delta \boldsymbol{v}\) determines the direction of acceleration \( \boldsymbol{a} \). Consequently, \( \boldsymbol{a} \) also points to the center of the circle. The acceleration vector \( \boldsymbol{a} \) is perpendicular to the velocity vector \( \boldsymbol{v} \) at every point of the circular path. Let's add \( \boldsymbol{a} \) with the small index \(\text z\): \( \boldsymbol{a}_{\text z} \) to indicate that it is centripetal acceleration ("centripetal" = "acting towards the center of the circle"). \( \boldsymbol{a}_{\text z} \) is sometimes also called radial acceleration ("radial" = "along the radius").

We have figured out the direction of the centripetal acceleration. Now we have to find out its amount \( a_{\text z} \).

When the body moved from \(S_1\) to \(S_2\), it covered the angle \(\Delta \varphi\). \(\Delta \varphi\) is the angle between the straight lines \(C\,S_1\) and \(C\,S_2\) (see illustration 1). It can also be easily seen that the angle \(\delta \theta\) between \( \boldsymbol{v}_1 \) and \( \boldsymbol{v}_2 \) is equal to the angle \(\delta \varphi \): \(\Delta \theta ~=~ \Delta \varphi \) (see illustration 2).

We now exploit the trigonometric relationship for sines. The sine of an angle is equal to the opposite cathetus divided by the hypotenuse. From the blue triangle (see illustration 2) you can read the following:

You can read off from the green triangle:

If we just equate Eq. 7 and Eq. 8, we eliminate the angle:

Divide both sides in Eq. 9 by \(\Delta t\):

In this way you bring the centripetal acceleration \(a_{\text z} = \Delta v / \Delta t\) into play. Let's take advantage of the fact that \( \Delta s / \Delta t\) (distance per time) is the orbital velocity \(v\):

Rearrange Eq. 11 with respect to the centripetal acceleration:

So e can say: If a body moves on a circular path with radius \(r\), with a constant velocity \(v\), then the body is accelerated towards the center of the circle.

We can also express the velocity \(v\) with the period \(T\) and the frequency \(f\) of the oscillation. Period \(T\) is the duration of one revolution and frequency \(f\) is the number of revolutions per second. The two quantities are related as follows:

The circumference is given by \(2\pi r\). This distance is covered within the period \(T\) because \(T\) is the duration of one revolution. The velocity \(v\) is then given as distance per time (here circumference per time):

The centripetal acceleration can also be expressed with the frequency \(f\):

Or alternatively with the circular frequency \(\omega = 2\pi \, f \):