imaginary line segment extending from the center of a circle to the particle. To use this method,
one also needs to define a reference line segment—the positive x axis is the conventional choice
for the case of a circle centered on the origin of an x-y coordinate system. Then, as long as you
know the radius r of the circle, the angle θ that the line to the particle makes with the reference
line completely specifies the location of the particle.
v
r
s
θ
In geometry, the position variable s, defines an arc length on the circle. Recall that, by
definition, the angle θ in radians is the ratio of the arc length to the radius:
s
θ =
r
Solving for s we have:
s = r θ
(18-1)
in which we interpret the s to be the position-on-the-circle of the particle and the θ to be the
angle that an imaginary line segment, from the center of the circle to the particle, makes with a
reference line segment, such as the positive x-axis. Clearly, the faster the particle is moving, the
faster the angle theta is changing, and indeed we can get a relation between the speed of the
particle and the rate of change of θ just by taking the time derivative of both sides of equation
18-1. Let’s do that.
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