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 2dimensional.
(XB,YB)
r
(XA,YA)
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 Chapter 9 when we study 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.
C.
Acceleration
1.
Acceleration is defined as the
Time Rate of Change of the Velocity
Vector.
A
Velocity Vector has two parts: Magnitude (Speed) and Direction.
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
This is the acceleration an object feels due to a change in the object’s
Speed.
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
Same as Velocity if Speeding Up
Opposite of Velocity if Slowing Down
Speeding Up
3.
Slowing Down
Centripetal (Radial) Acceleration
Centripetal means
Center Seeking
This tells you that the centripetal acceleration always points to the
the
Center of
Circle. It is therefore Perpendicular to the Tangential
Acceleration.
Centripetal acceleration is due to the change in the
Velocity Vector.
Direction of the
Any object traveling in a
Curved Path MUST HAVE Centripetal
Acceleration.
Furthermore, notice that the
Centripetal Acceleration is always
Perpendicular to the Tangential Acceleration.
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:
v2
a   2 r
r
This is a very useful formula for solving problems and can be derived directly from the
definition of acceleration using Calculus. This derivation is usually reserved for students in either
Engineering Principles I (Dynamics) or the Junior Level Mechanics class for Physics and
Engineering Physics Majors. You will probably prefer just to memorize the equation.
4.
Total Acceleration
The total acceleration of an object traveling in a circle is 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
acceleration.
III.
It does have
no tangential
centripetal acceleration.
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!!
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