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

Chapter 18 Circular Motion: Centripetal Acceleration

How the Centripetal Acceleration Depends on the Speed of the Particle and

the Size of the Circle

We are now in a position to derive an expression for that center-directed (centripetal)

acceleration we were talking about at the start of this chapter. Consider a short time interval ∆t.

(We will take the limit as ∆t goes to zero before the end of this chapter.) During that short time

interval, the particle travels a distance ∆s along the circle and the angle that the line, from the

center of the circle to the particle, makes with the reference line changes by an amount ∆θ.

v′

v

∆s

r

∆θ

s

θ

Furthermore, in that time ∆t, the velocity of the particle changes from v to v ′ , a change v

∆

defined by v ′ = v + v

∆ depicted in the following vector diagram (in which the arrows

representing the vectors v and v ′ have been copied from above with no change in orientation or

length). Note that the small angle ∆θ appearing in the vector addition diagram is the same ∆θ

that appears in the diagram above.

∆v

v

∆θ

v′

1

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