Studies related to trigonometric arcs have applications in the context of Physics, especially in situations involving circular movements. In Physics, some bodies develop circular trajectories, so they travel through spaces at certain times, have angular velocity and acceleration.
Let's consider a rover on a circular path of radius R and center C, counterclockwise, considering O the origin of the spaces and P the position of the rover at a given time. See illustration:
Let's determine the angular space (φ) and the average angular velocity (ωm) of the mobile.
Angular space (φ)
It is given by the opening of vertex C, corresponding to the OP path arc. In this case OP is the space s and the angle φ is given in radians (rad).
Average angular velocity (ωm)
It is the relationship between the variation in angular space (∆φ = φ 2 – φ1) and the variation in the time taken to travel through space (∆t = t2 – t1).
Example 1
A point traverses a circular region and describes a central angle of 2 rad in 5 seconds. Determine the average angular velocity over this time interval.
Data:
central angle: φ = 2 rad
time: ∆t = 5 seconds
ωm = 2/5 → ωm = 0.4 rad/s
Example 2
Determine the time interval that a rover takes to traverse the arc of circumference AB, indicated in the figure, with a constant scalar velocity equal to 24m/s.
1st step: determine the space between A and B
s = φ * R
s = 3 * 160
s = 480 m
2nd step: determine the time spent
by Mark Noah
Graduated in Mathematics
Brazil School Team
Trigonometry - Math - Brazil School
Source: Brazil School - https://brasilescola.uol.com.br/matematica/arcos-movimento-circular.htm