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

$$\overline{)\phi ={\phi}_{0}+\omega \xb7t}$$

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:

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*:

$$\mathrm{tan}\theta =\frac{\text{opposite\hspace{0.17em}cathetus}}{\text{adjacentcathetus}}=\frac{\u2206\phi}{\u2206t}=\frac{\phi -{\phi}_{0}}{t}=\omega $$

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

$$\overline{)\omega ={\omega}_{0}=\text{cte}}$$

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:

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 - φ_{0}).

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

$$\overline{)\alpha =0}$$

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/s*), 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:

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