Lesson 1: Motion
Characteristics for Circular Motion

Acceleration

As mentioned earlier in Lesson
1, an object moving in uniform circular motion is moving
in a circle with a uniform or constant speed. The velocity
vector is constant in magnitude but changing in direction.
Because the speed is constant for such a motion, many
students have the misconception that there is no
acceleration. "After all," they might say, "if I were
driving a car in a circle at a constant speed of 20 mi/hr,
then the speed is neither decreasing nor increasing;
therefore there must not be an acceleration." At the center
of this common student misconception is the wrong belief
that acceleration has to do with speed and not with
velocity. But the fact is that an
accelerating object is an object which is changing its
velocity. And since velocity
is a vector which has both magnitude and direction, a
change in either the magnitude or the direction constitutes
a change in the velocity. For this reason, it can be safely
concluded that an object moving in a circle at constant
speed is indeed accelerating. It is accelerating because the
direction of the velocity vector is changing.

To understand this at a deeper level, we
will have to combine the definition of acceleration with a
review of some basic vector principles. Recall from
Unit 1 of The Physics
Classroom that acceleration as a quantity was defined as
the rate at which the velocity of an object changes. As
such, it is calculated using the following equation:

where v_{i}
represents the initial velocity and
v_{f} represents
the final velocity after some time of
t. The numerator of the
equation is found by subtracting one vector
(v_{i}) from a
second vector
(v_{f}). But the
addition and subtraction of vectors from each other is done
in a manner much different than the addition and subtraction
of scalar quantities. Consider the case of an object moving
in a circle about point C as shown in the diagram below. In
a time of t seconds, the object has moved from point A to
point B. In this time, the velocity has changed from
v_{i} to
v_{f}. The
process of subtracting
v_{i} from
v_{f} is shown
in the vector diagram; this process yields the change in
velocity.

Direction of the
Acceleration Vector

Note in the diagram above that there is a velocity change
for an object moving in a circle with a constant speed. A
careful inspection of the velocity change vector in the
above diagram shows that it points down and to the left. At
the midpoint along the arc connecting points A and B, the
velocity change is directed towards point C - the center of
the circle. The acceleration of the object is dependent upon
this velocity change and is in the same direction as this
velocity change. The acceleration of the object is in the
same direction as the velocity change vector; the
acceleration is directed towards point C as well - the
center of the circle. Objects moving in circles at a
constant speed accelerate towards the center of the
circle.

The
acceleration of an object is often measured using a device
known as an accelerometer. A simple accelerometer consists
of an object immersed in a fluid such as water. Consider a
sealed jar which is filled with water. A cork attached to
the lid by a string can serve as an accelerometer. To test
the direction of acceleration for an object moving in a
circle, the jar can be inverted and attached to the end of a
short section of a wooden 2x4. A second accelerometer
constructed in the same manner can be attached to the
opposite end of the 2x4. If the 2x4 and accelerometers are
clamped to a rotating platform and spun in a circle, the
direction of the acceleration can be clearly seen by the
direction of lean of the corks. As the cork-water
combination spins in a circle, the cork leans towards the
center of the circle. The least massive of the two objects
always leans in the direction of the acceleration. In the
case of the cork and the water, the cork is least massive
(on a per mL basis) and thus it experiences the greater
acceleration. Having less inertia (owing to its smaller mass
on a per mL basis), the cork resists the acceleration the
least and thus leans to the inside of the jar towards
the center of the circle. This is observable evidence that
an object moving in circular motion at constant speed
experiences an acceleration which is directed towards the
center of the circle.

Another simple homemade accelerometer involves a lit
candle centered vertically in the middle of an open-air
glass. If the glass is held level and at rest (such that
there is no acceleration), then the candle flame extends in
an upward direction. However, if you hold the glass-candle
system with an outstretched arm and spin in a circle at a
constant rate (such that the flame experiences an
acceleration), then the candle flame will no longer extend
vertically upwards. Instead the flame deflects from its
upright position. This signifies that there is an
acceleration when the flame moves in a circular path at
constant speed. The deflection of the flame will be in the
direction of the acceleration. This can be explained by
asserting that the hot gases of the flame are less massive
(on a per mL basis) and thus have less inertia than the
cooler gases which surround. Subsequently, the hotter and
lighter gases of the flame experience the greater
acceleration and will lean in the direction of the
acceleration. A careful examination of the flame reveals
that the flame will point towards the center of the circle,
thus indicating that not only is there an acceleration; but
that there is an inward acceleration. This is one more piece
of observable evidence which indicates that objects moving
in a circle at a constant speed experience an acceleration
which is directed towards the center of the circle.

So
thus far, we have seen a geometric proof and two real-world
demonstrations of this inward acceleration. At this point it
becomes the decision of the student to believe or to not
believe. Is it sensible that an object moving in a circle
experiences an acceleration which is directed towards the
center of the circle? Can you think of a logical reason to
believe in say no acceleration or even an outward
acceleration experienced by an object moving in uniform
circular motion? In the next part of
Lesson 1, additional logical evidence will be presented
to support the notion of an inward force for an object
moving in circular motion.

Check
Your Understanding

1. The initial and final speed of a ball at two different
points in time is shown below. The direction of the ball is
indicated by the arrow. For each case, indicate if there is
an acceleration. Explain why or why not. Indicate the
direction of the acceleration.

a.

Acceleration: Yes or No? Explain.

If there is an acceleration, then what
direction is it?

b.

Acceleration: Yes or No? Explain.

If there is an acceleration, then what
direction is it?

c.

Acceleration: Yes or No? Explain.

If there is an acceleration, then what
direction is it?

d.

Acceleration: Yes or No? Explain.

If there is an acceleration, then what
direction is it?

e.

Acceleration: Yes or No? Explain.

If there is an acceleration, then what
direction is it?

2. Explain the connection between your answers to the
above questions and the reasoning used to explain why an
object moving in a circle at constant speed can be said to
experience an acceleration.

3. Dizzy Smith and Hector Vector are still discussing
#1e. Dizzy says that the ball is not accelerating because
its velocity is not changing. Hector says that since the
ball has changed its direction, there is an acceleration.
Who do you agree with? Argue a position by explaining the
discrepancy in the other student's argument.

4. Identify the three controls on an automobile which
allow the car to be accelerated.

For questions #5-#8:
An object is moving in a clockwise direction around a
circle at constant speed. Use your understanding of the
concepts of velocity and acceleration to
answer the next four questions. Use the diagram shown at the
right.

5. Which vector below represents the direction of the
velocity vector when the object is located at point B on the
circle?

6. Which vector below represents the direction of the
acceleration vector when the object is located at point C on
the circle?

7. Which vector below represents the direction of the
velocity vector when the object is located at point C on the
circle?

8. Which vector below represents the direction of the
acceleration vector when the object is located at point A on
the circle?

Lesson 1: Motion
Characteristics for Circular Motion