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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