Circular Motion
I.
Circular Motion and Polar Coordinates
A.
Consider the motion of ball on a circle from point A to point B as
shown below. We could describe the path of the ball in Cartesian
coordinates or by polar coordinates. In Cartesian coordinate
system, we see that both coordinates change!! This makes the
problem 2-dimensional.
(XB,YB)
(XA,YA)
r
B
If we use polar coordinates, the radius is constant and only the angle
theta changes. This simplifies the system to a 1-dimensional
problem and makes the math simpler.
We will deal with this in more detail in the Chapter on rotation!
B.
Tangential Velocity
By its definition in terms of the derivative of the position vector, the
velocity vector is Tangent to the Curve every point on the ball's path.
Thus, we call it the Tangential velocity.
This is the same velocity that we dealt with in 1-dimensional and
projectile motion problems.
For a rigid body composed of many particles traveling in circles of
different radii, it is convenient to also define another type of velocity
(angular velocity) based upon polar coordinates. Thus, we need to be
careful in our terminology when discussing velocity to ensure the
reader knows which velocity we are talking about.
C.
Acceleration
1.
Acceleration is defined as the Time Rate of Change of the Velocity
Vector.
A Velocity Vector has two parts: ___________ and ____________.
If either part changes then the object is undergoing Acceleration.
Thus, it is often convenient to break the acceleration into
components based upon the change in speed (magnitude) or
direction of the velocity vector instead of x and y directions. This is
again an example of using polar coordinates to represent the motion.
2.
Tangential Acceleration
Tangential acceleration is the acceleration an object feels due to a
change in the object’s ________________.
The magnitude of the tangential acceleration is usually either
specified in the problem statement or found using trigonometry.
The tangential acceleration is the only acceleration possible for
straight line motion. We can use this to help us find the direction of
the acceleration vector.
Direction of Tangential Acceleration
_______________ as Velocity vector if object is Speeding Up
_______________ of Velocity vector if object is Slowing Down
Speeding Up
3.
Slowing Down
Centripetal (Radial) Acceleration
Centripetal acceleration is due to the change in the
______________ of the Velocity Vector
Centripetal means Center Seeking- This tells you that the centripetal
acceleration always points to the Center of the Circle.
It is always Perpendicular to the Tangential Acceleration.
Any object traveling in a ____________Path MUST HAVE
Centripetal Acceleration.
Furthermore, notice that the Centripetal Acceleration is always
Perpendicular to the Tangential Velocity Vector.
This is why the moon can be accelerating toward the Earth while not
moving toward the Earth!!
The magnitude of the centripetal acceleration vector can be found
by the formula:
a
v2
 2 r
r
This is a very useful formula for solving problems. Students in trig.
based physics courses can either derive the equation using
trigonometry or just memorize the equation. However, the equation
can be derived easily using the basic definition of acceleration
using polar coordinates once a student takes Calculus.
4.
Total Acceleration
Tangential and Centripetal Acceleration are our acceleration
components in polar coordinates. The total acceleration of an
object traveling in a circle is thus the vector sum of the tangential
acceleration and the centripetal acceleration.
Example: A car is slowing down at a rate of 6.00 m/s2 while
traveling counter clockwise on a circular track of radius 100.0 m.
What is the total acceleration on the car when it has slowed to 20.0
m/s as shown below:
II.
Uniform Circular Motion
An object that is traveling in a circle at constant speed is said to be
traveling in uniform circular motion.
This is just a special case of circular motion where the object has no
tangential acceleration. It does have centripetal acceleration.
Concept Question
Consider the case of projectile motion from the last lesson: A cannon ball is
fired out of a cannon and follows a parabolic path before hitting the ground.
What type(s) of acceleration does the cannon ball have during its flight at
point X?
X
A.
Tangential Acceleration
B.
Centripetal Acceleration
C.
Both Tangential and Centripetal Acceleration
D.
Neither Centripetal or Tangential Acceleration
III. General Curve-linear Motion In A Plane
As our concept question shows, any curve-linear motion can be seen
at every instant as circular motion a circle whose radius is the radius
of curvature of the trajectory at that particular point. In the case of
straight line motion, the radius of curvature is infinity so their is no
centripetal acceleration. In the case of circular motion, the radius of
curvature is constant!
So why didn't we treat projectile motion using the concepts of
centripetal and tangential acceleration?
Because it makes the math harder to perform!! In Cartesian
coordinates, the acceleration has only one component (vertical) and it
is constant in magnitude. Thus, we can use the kinematic equations.
In polar form, both the tangential and centripetal acceleration
components vary in direction and magnitude. Thus, we couldn't use
the kinematic equations with either component.
We Use Different Coordinate Systems and Define New Quantities To
Make The Math Simpler For Solving Problems. This Doesn't Mean
That There Is New Physics!!