# Lesson 8

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```					                                            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)

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                                               Slowing Down

Centripetal means   Center Seeking

This tells you that the centripetal acceleration always points to the   Center of

the   Circle. It is therefore Perpendicular to the Tangential

Acceleration.

Centripetal acceleration is due to the change in the   Direction of the
Velocity Vector.
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   no tangential
acceleration.           It does have   centripetal 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!!
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

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