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Circular Motion
Kinematics of Uniform Circular Motion
Dynamics of Uniform Circular Motion
Car Rounding a Curve
Chapter 5
Circular Motion
Gravitation
Motion in Chapter 4 was always in a straight line. In this
chapter, we introduce circular motion.
Newton’s laws still apply. New chapter, old problems.
The biggest difference involves the acceleration…
We’ll get to gravitation after exam 2.
5.1 Kinematics of Uniform Circular Motion
Do you remember the equations of kinematics?
There are analogous equations for rotational quantities. You
will see them later in the course. This section is somewhat
mis-titled. I believe our starting point for circular motion best
involves forces (dynamics).
Consider an object moving in a straight line.
Vi
A force F applied parallel to the
direction of motion for a time ∆t
increases the magnitude of velocity
by an amount a∆t, but does not
change the direction of motion.
F
Vi
∆V=a∆t
A force applied parallel to an object’s velocity vector increases
the object’s speed.
A force F applied perpendicular to
the direction of motion for a time
∆t changes the direction of the
velocity vector.
F
Vi
∆V=a∆t
In the limit ∆t→0, the length of the velocity vector does not
change.
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A force applied perpendicular
to an object’s velocity vector
instantaneously changes the
direction of the velocity vector,
but not the object’s speed.
Summary and consequences:
F
A force applied parallel to an object’s velocity vector increases
the object’s speed.
Vf
Vi
∆V=a∆t
If the applied force is always perpendicular to the velocity
vector, the object constantly changes direction, but never
speeds up or slows down.
A force applied perpendicular to an object’s velocity vector
instantaneously changes the direction of the velocity vector,
but not the object’s speed.
If you apply a force parallel to the velocity vector you can only
change an object’s speed, not its direction of motion.
The object follows a circular path!
If you apply a force perpendicular to an object’s velocity
vector, you will change its direction of motion BUT NOT ITS
SPEED!
An example of the latter is circular motion.
Circular motion “toys:”
• A ball tied to the end of a string and “whirled” around.
Position, velocity, and acceleration in uniform circular
motion.
• A child on a merry-go-round.
Circular motion review.
• A car rounding a circular curve.
Car rounding a bend.
• The earth orbiting the sun (approximately).
Car on a circular banked track (click the “play”
button, then click “initialize/reset.”
• The moon orbiting the earth (approximately).
Hot Wheels Track:
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An object moving in a circle with constant speed is said to
undergo uniform circular motion.
A ball on a string:
The force that accelerates the ball is the tension in the string
to which it is attached.
v
The instantaneous velocity is tangent to
the path of motion (OK to “attach” velocity
to object—this is not a free-body diagram).
The centripetal force due to the string
gives rise to a centripetal (also called
radial) acceleration.
a
The instantaneous acceleration is
perpendicular to the velocity vector.
If the ball moves uniformly in a circle, both
the force and acceleration continually
change direction, so that they always point
to the center of the circle.
The ball is accelerated because its velocity constantly changes.
If the motion is uniform circular, the acceleration is towards
the center of the circle; i.e., the acceleration is “radial” or
“centripetal.”
Your book shows that
Note: if the motion is not uniform (the speed changes or the
radius of the circle changes) there will also be a tangential
acceleration. We will not worry about that case in Physics
31.
Circular motion: ball on a string, horizontal circle.
OSE :
ar =
T
ac
v2
r
For the problems from this chapter, you need to recall the
definitions of frequency, period, and know how to use the
fact that that an object moving in a circle with constant speed
has speed given by v = 2πr / T. Note that this is not on your
OSE sheet. Perhaps it should be.
circumference = 2πr
speed =
distance
time
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5.2 Dynamics of Uniform Circular Motion
Newton’s Laws still apply.
The OSE sheet contains a variation on Newton’s second law
∑F
OSE :
r
= m ar
where the subscript “r” stands for “radial.” You may also use the
subscript “c” (“centripetal”).
Example. Suppose the ball (mass m) in the example I gave in
the previous sections moves with a constant speed V. What is
the tension in the string?
Choose an axis parallel to the
instantaneous acceleration vector. The
direction of the y-axis is irrelevant here.
∑F
x
= m ax
x
NO!
V
T
ac
“How about a velocity force?”
NO, NO, NO!
T
ac
Tx = m ax
 V2 
+T
=
m
( )
+

 r 
T =m
“Isn’t there an outward force, pulling the ball out?”
x
V2
r
“What about the centrifugal force I feel when my car goes
around a curve at high speed?”
There is no such thing* as centrifugal force!
“But I feel a force!”
You feel the centripetal force of the door
pushing you towards the center of the circle of
the turn!
Your confused brain interprets the effect of Newton’s first law
as a force pushing you outward.
*Engineers talk about centrifugal force when they attach a coordinate system to an
accelerated object. In order to make Newton’s Laws work, they have to invent a
pseudoforce, which they call “centrifugal force.”
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Example 5-2. The moon’s nearly circular orbit about the
Earth has a radius of 384,000 km and a period T of 27.3 days.
Determine the acceleration of the moon toward the earth.
I’ll work this example if there is time.
5.3 A Car Rounding a Curve
When you are riding in a car rounding a sharp curve, you may
think you feel a centrifugal force.
In reality, your body “wants” to continue in a straight line, and
you feel the seat force accelerating you along the curved path
the car follows.
There is a web page for this text, with lots of good material.
Here is the Chapter 5 starting page.
But first—this may seem strange, but the part of a rolling tire
that is in contact with the road is instantaneously at rest.
The friction force between rolling tire and road is thus a static
friction force.
We can use our expert techniques and equation for centripetal
force to solve car-rounding-curve problems.
Example. Calculate the minimum coefficient of friction
between tires and road that will allow a car traveling with a
speed V to round a flat curve of radius R without sliding.
We’ll explore this in more detail later.
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Example 5-8. For a car traveling with speed V around a
curve of radius R, determine the angle at which the road
should be banked so no friction is required.
Example. What is the maximum safe speed around a banked
curve of radius R and incline θ with road coefficient of friction
µ?
See Physics 23 Lecture 8.
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