Calculus-Based Physics by Jeffrey W. Schnick - HTML preview

Chapter 18 Circular Motion: Centripetal Acceleration

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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