While v ′ is a new vector, different from v , we have stipulated that the speed of the particle is a
constant, so the vector v ′ has the same magnitude as the vector v . That is, v ′ =v . We redraw
the vector addition diagram labeling both velocity vectors with the same symbol v .
∆v
v
∆θ
v
The magnitude of the centripetal acceleration, by definition, can be expressed as
∆v
a = lim
c
∆t→0 ∆t
Look at the triangle in the vector addition diagram above. It is an isosceles triangle. The two
unlabeled angles in the triangle are equal to each other. Furthermore, in the limit as ∆t
approaches 0, ∆θ approaches 0, and as ∆θ approaches 0, the other two angles must each
approach 90° in order for the sum of the angles to remain 180°, as it must, because the sum of
the interior angles for any triangle is 180°. Thus in the limit as ∆t approaches 0, the triangle is a
right triangle and in that limit we can write:
∆v = tan( θ
∆ )
v
∆v =v
(
tan θ
∆ )
Substituting this into our expression for ac we have:
v tan ∆
( θ )
a = lim
(18-4)
c
∆t→ 0
∆t
Now we invoke the small angle approximation from the mathematics of plane geometry, an
approximation which becomes an actual equation in the limit as ∆θ approaches zero.
The Small Angle Approximation
For any angle that is very small compared to π radians (the smaller the angle the better
the approximation), the tangent of the angle is approximately equal to the angle itself,
expressed in radians; and the sine of the angle is approximately equal to the angle itself,
expressed in radians. In fact,
tan( θ
∆ )
→ θ
∆ , and sin( θ
∆ )
→ θ
∆
θ
∆ →0
θ
∆ →0
where θ
∆ is in radians.
The small angle approximation allows us to write
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