Review Sheet by Leon Lin
Chapter 6:
Circular Motion and Other Application of Newton’s Laws
Summary:
This chapter introduces the application of Newton’s laws to uniform circular motion.

What is Uniform Circular Motion?
o Uniform circular motion is when an object
moves around a circular path (such as a rocket
ship circling the Earth or a rock in a sling being
spun around) at a constant speed.

Constant Speed?
o The object is moving a set distance in the circle
for each unit of time that passes. NOTE THAT
THIS DOES NOT MEAN THE OBJECT HAS A
CONSTANT VELOCITY. The object may be
moving at the same speed but speed is a
SCALAR quantity, while velocity is a VECTOR
quantity. The speed may not be changing but
the direction the particle is object is moving
certainly is.

So how do we apply Newton’s Laws to Uniform
Circular Motion?
o According to the book, “Newton’s second law
applied to a particle moving in uniform circular
motion states that the net force causing the
particle to undergo a centripetal acceleration is:
This is a vertical uniform circular
motion example. Roller coaster loops
are one of the most classic examples of
vertical uniform circular motion


o
This is a horizontal uniform circular motion
example. An object rotating on a string is
one of the most classic examples of
horizontal uniform circular motion

As you can see, the centripetal acceleration of
any particle moving in a circular motion is equal
to the velocity the object is moving at divided by
the radius of the circle.
Pitfalls and Misconceptions
o The centripetal force pulls objects towards the
center of its circular motion. The commonly
heard centrifugal force is an IMAGINARY force
that pulls things away from the center of the
circle and does not actually exist.
Example Problems!
Let’s start with something easy:
An 800 kg space satellite is orbiting the earth at a constant speed of 7000 m/s in a circular path
of 265,000 km long around the Earth. Find the satellite’s centripetal acceleration and the net
centripetal force responsible for its motion.
Solution to Easy:
Well, we know the satellite is moving at a constant speed hence we should use the above
equation of
to solve the equation. But first, let’s convert everything to the same
unit. Since the numbers are all so large, I’m going to convert everything to km.
So now we have m = 800kg, v = 7 km/s, and r = 265,000 km
To find acceleration we use the equation
(
, so just plug and chug:
)
Now to find the rest, we just use the new centripetal acceleration we found to calculate the total
centripetal force:
Now for something slightly harder:
θ
A race car is travelling around a banked course (radius of r) at velocity v. At what angle θ should
we bank the course so that the car can travel smoothly along the course with no friction from the
tires?
Solutions to Medium:
Well, we know that we will have to tilt our axes for this problem. So let’s set up what our forces
will be:
Force Normal X
Force Normal Y
Force Normal
Normal
Force
Force of Gravity
As we can see
and
. Well this way
we can cancel out the two force normals by dividing them over one another like this:
The masses cancel to give you:
Simply take the inverse tangent of
to get the angle.
Ready for a challenge?
A roller coaster car enters the loop shown above. At the top of the circle, the speed of the car is
v m/s and the acceleration points straight down. If the diameter of the loop is d m and the weight
of the car and the passengers is m kg, find the magnitude of the normal force exerted by the
track on the car at this point. Also find the normal force exerted by the track on the car when it is
at the bottom of the loop assuming the speed remains constant.
Solution to Hard:
When looking at circular motion problems, think of all the forces pointing toward the center as
positive and any pointing away from the center of the circle to be negative. This way, there are
two forces acting on the car at its topmost point: the normal force and the gravitational force are
both pointing downwards TOWARDS the center, hence both are positive.
So a force analysis would show:
at the top of the loop.
Now for the bottom:
at the bottom of the loop.