176

9.2Uniform Circular Motion

In this section I derive a simple and very useful equation for the magni-

tude of the acceleration of an object undergoing constant acceleration. The

law of sines is involved, so I’ve recapped it on the left.

The derivation is brief, but the method requires some explanation and

justification. The idea is to calculate a

.

v vector describing the change in

the velocity vector as the object passes through an angle

.

. We then calcu-

late the acceleration, a=

.

v/

.

t. The astute reader will recall, however, that

this equation is only valid for motion with constant acceleration. Although

the magnitude of the acceleration is constant for uniform circular motion,

the acceleration vector changes its direction, so it is not a constant vector,

and the equation a=

.

v/

.

t does not apply. The justification for using it is

that we will then examine its behavior when we make the time interval very

short, which means making the angle

.

very small. For smaller and smaller

time intervals, the

.

v/

.

t expression becomes a better and better approxima-

tion, so that the final result of the derivation is exact.

In figure (a), the object sweeps out an angle

.

. Its direction of motion

also twists around by an angle

.

, from the vertical dashed line to the tilted

one. Figure (b) shows the initial and final velocity vectors, which have equal

magnitude, but directions differing by

.

. In (c), the vectors have been

reassembled in the proper orientation for vector subtraction. They form an

isosceles triangle with interior angles

.

,

.

, and

.

. (Eta,

.

, is my favorite

Greek letter.) The law of sines gives

.

v

sin

.

=v

sin

.

.

This tells us the magnitude of

.

v, which is one of the two ingredients we

need for calculating the magnitude of a=

.

v/

.

t. The other ingredient is

.

t.

The time required for the object to move through the angle

.

is

.

t =

lengthofarc

v

.

Now if we measure our angles in radians we can use the definition of radian

measure, which is (angle)=(length of arc)/(radius), giving

.

t=

.

r/|v|. Com-

bining this with the first expression involving |

.

v| gives

|a|= |

.

v|/

.

t

=

v

2

r

·

sin

.

.

·

1

sin

.

.

When

.

becomes very small, the small-angle approximation sin

.»

applies, and also

.

becomes close to 90

°

, so sin

.˜

1, and we have an

equation for |a|:

|a| =

v

2

r

[uniform circular motion] .

.

.

(a)

A

B

C

a

b

c

The law of sines:

A/sin a = B/sin b = C/sin c

(b)(c)

v

i

v

f

-v

i

v

f

.

v = v

f

+ (-v

i

)

.

..

Chapter 9Circular Motion