A body performs a uniform circular motion (u.c.m.) when its trajectory is a circumference and its angular velocity is constant. In this section, we are going to study the following graphs:

U.C.M. Graphs

Angular position-time (φ-t) graph


The angular position–time graph (φ–t) of a uniform circular motion (u.c.m.). represents time on the horizontal axis (t–axis) and angular position on the vertical axis (φ-axis). The angular position, φ, measured in radians according to the units of the international system (S.I.), increases (or decreases) in a uniform manner with the passage of time. We can distinguish two cases, according to whether the angular velocity is positive or negative:

Angular position - time graph in uniform circular motion.

From the angle α, we can get the angular velocity ω. To do it just remember that in a right triangle the tangent of each of its angles is defined as the opposite cathetus divided by the adjacent one:

tan θ=opposite cathetusadjacent cathetus=φt=φ-φ0t=ω

The value of the slope is the value of the angular velocity ω. So, the greater the slope of the straight line, the higher angular velocity ω of the body.

Angular velocity - time (ω-t) graph


The angular velocity-time graph (ω–t) of a uniform circular motion (u.c.m.) shows that the angular velocity ω, measured in radians per second (rad/s) according to the international system (S.I.), remains constant over time. Again, we can distinguish two cases:

Angular velocity - time graph in uniform circular motion.

Notice that the area under the curve ω between two instants of time is the angular distance traveled, that is, the angle covered: ∆φ = φ - φ0).

Angular distance traveled in uniform circular motion.

In this case, it is just the calculation of said area, since it is a rectangle. But, do you know what mathematical tool enables the calculation of the area under a curve, whatever its form?

Angular acceleration - time (α-t) graph


The angular acceleration-time graph (α-t) of a uniform circular motion (u.c.m.) shows that the angular acceleration, measured in the international system (S.I.) in radians per second squared (rad/s2), is zero at all times. In this case, whether the velocity of the body is considered to be positive or negative, there is only one possibility, as illustrated in the figure:

Angular acceleration - time in uniform circular motion

Solved exercises worksheet

Here you can test what you have learned in this section.

s–t, v–t, at–t and an–t graphs in u.c.m.


Based on the graphs of angular position–time (φ–t), angular velocity–time (ω–t), angular acceleration–time (α–t) of uniform circular motion (u.c.m.) draw the graphs of distance traveled–time (s–t), velocity–time (v–t), tangential acceleration–time (at–t) and normal acceleration–time (an–t). Considering that angular velocity is ω>0.

Formulas worksheet

Here is a full list of formulas for the section Uniform Circular Motion (U.C.M.) Graphs. By understanding each equation, you will be able to solve any problem that you may encounter at this level.

Click on the icon   to export them to any compatible external program.

Angular position in u.c.m.


Angular velocity in u.c.m.


Angular acceleration (u.c.m.)


Definition of tangent of an angle

tanα=opposite cathetusadjacent cathetus=bc

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