Unizor - Physics4Teens - Mechanics - Kinematics - Rotation
Channel: Zor Shekhtman on UNIZOR Education
Date: 2018-04-23

Unizor - Physics through a Looking Glass of Mathematician


Let's consider a rotation of an object within XY-plane around Z-axis along a circular trajectory of radius R with a center at the origin of coordinate.
Obviously, its Z-coordinate will always be zero, that is z(t)=0, but X- and Y-coordinates will somehow depend on time t.

We would like this motion to be uniform in some sense, but we have to define what we really mean calling this motion a uniform rotation.
Recall that the term uniform motion we used in a sense that an object moves along a straight line with constant velocity, covering the same distance in the same intervals of time.
When dealing with rotation, the concept of distance along a straight line should be replaced with a concept of angle of rotation. Inasmuch as we cover the same linear distance in uniform motion during the same intervals of time, in case of uniform rotation we cover the same angle of rotation during the same intervals of time.

Assume that in the beginning of rotation our object is at point {R, 0, 0} - on the X-axis at distance R from the origin of coordinates. Let's introduce a new variable - angle of rotation as a function of time - φ(t) - an angle from the positive direction of the X-axis counterclockwise to a position vector of our object at time t.
Since during the rotation along a circular trajectory the distance between an object and the origin of coordinates is constant R, we can express coordinate functions as functions of an angle of rotation, like in polar coordinates:
x(t) = R·cos(φ(t))
y(t) = R·sin(φ(t))

Since we are analyzing the uniform rotation, and this uniformity means that we rotate by the same angle during the same intervals of time, we define this motion as a rotation with constant angular speed φ'(t) - constant first derivative of an angle of rotation by time.
Assume, angular speed φ'(t)=ω (constant).
Then angle of rotation φ(t) = ω·t + φ0, where φ0 is an angle from the positive direction of the X-axis to a vector from the origin of coordinates to a position of our object at the beginning of motion at t=0. Since we assumed that the object starts from a point {R, 0, 0} on the X-axis, φ0=0 and the angle of rotation equals to φ(t) = ω·t.

Now we can easily find the coordinate functions of the position of an object:
x(t) = R·cos(ω·t)
y(t) = R·sin(ω·t)
z(t) = 0