Lectures on Physics has been derived from Benjamin Crowell's Light and Matter series of free introductory textbooks on physics. See the editorial for more information....

In this section I derive a simple and very useful equation for the
magnitude of the acceleration of an object undergoing constant acceleration.
The law of sines is involved, so I've recapped it in figure
f.

f / The law of sines.

The derivation is brief, but the method requires some explanation
and justification. The idea is to calculate a Δv vector describing
the change in the velocity vector as the object passes through an
angle θ. We then calculate the acceleration, a = Δv/Δt. 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 = Δv/Δt 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 θ very small. For
smaller and smaller time intervals, the Δv/Δt expression becomes
a better and better approximation, so that the final result of the
derivation is exact.

g / Deriving |a| = |v|^{2}/r for
uniform circular motion.

In figure g/1, the object sweeps out an angle θ. Its direction of
motion also twists around by an angle θ, from the vertical dashed
line to the tilted one. Figure g/2 shows the initial and final velocity
vectors, which have equal magnitude, but directions differing by θ.
In g/3, I've reassembled the vectors in the proper positions for vector
subtraction. They form an isosceles triangle with interior angles θ
,η, and η. (Eta, η, is my favorite Greek letter.) The law of sines
gives

This tells us the magnitude of Δv, which is one of the two ingredients
we need for calculating the magnitude of a = Δv/Δt. The other
ingredient is Δt. The time required for the object to move through
the angle θ is

Now if we measure our angles in radians we can use the definition of
radian measure, which is (angle) = (length of arc)/(radius), giving
Δt = θr/|v|. Combining this with the first expression involving
|Δv| gives

When θ becomes very small, the small-angle approximation sin θ ≈ θ
applies, and also η becomes close to 90 °, so sin η ≈ 1, and we have
an equation for |a|:

Force required to turn on a bike.

Don't hug the center line on a curve!

Acceleration related to radius and period of rotation.

A clothes dryer.

More about clothes dryers!.

→ Solved problem: The tilt-a-whirl page 231, problem 6

→ Solved problem: An off-ramp page 231, problem 7

Discussion Questions

A

A certain amount of force is needed to provide the acceleration of
circular motion. What if were are exerting a force perpendicular to the
direction of motion in an attempt to make an object trace a circle of radius
r , but the force isn't as big as m|v|^{2}/r?

B

i / Discussion question B. An
artist's conception of a rotating
space colony in the form of a giant
wheel. A person living in this
noninertial frame of reference has
an illusion of a force pulling her
outward, toward the deck, for the
same reason that a person in the
pickup truck has the illusion of
a force pulling the bowling ball.
By adjusting the speed of rotation,
the designers can make an
acceleration |v|^{2}/r equal to the
usual acceleration of gravity on
earth. On earth, your acceleration
standing on the ground is
zero, and a falling rock heads
for your feet with an acceleration
of 9.8 m/s^{2}. A person standing
on the deck of the space colony
has an upward acceleration of 9.8
m/s^{2}, and when she lets go of
a rock, her feet head up at the
nonaccelerating rock. To her, it
seems the same as true gravity.

Suppose a rotating space station, as in figure i on page 227, is built.
It gives its occupants the illusion of ordinary gravity. What happens when
a person in the station lets go of a ball? What happens when she throws
a ball straight "up" in the air (i.e., towards the center)?