CIRCULAR
MOTION
CIRCULAR MOTION

We will be looking at a special case of kinematics
and dynamics of objects in uniform circular
motion (constant speed)
Cars on a circular track (or on a curved road)
 Roller coasters in loop
 Other amusement park rides like ferris wheels, rotor,
scrambler
 Rotating objects (like a ball rolling) (these are moving
in a circular path even though radius is very small)
 Orbits of planets
 Running back cutting up field

Let’s determine the speed of the object.
Remember that speed is defined as:
v
d
t
(if speed is constant)
We define the period of motion (T) as the time it takes
to complete one rotation.
How far does it travel in one rotation?
We can find the circumference of the circular path by:
Therefore the speed of an object in
uniform circular motion is:
So the speed depends on the radius of the circle (think
about runners on a track – outside lane must run faster)
Ok so we’ve figured out its speed, but is it
accelerating?
Remember that it is traveling at a constant speed.
However, acceleration is defined as:
So how does the velocity change with respect to
time?
So even though speed is constant, velocity changes
WHICH DIRECTION IS VELOCITY?

http://www.youtube.com/watch?v=zww3IIMRo4U
Notice that the direction of the velocity at any time
is tangent to its path.
WHAT DIRECTION IS ACCELERATION
Acceleration is the change in velocity.
vf – vi – Let’s look at graphically
v
Acceleration is
towards center of
circle!
v
AN ANIMATION OF VELOCITY AND
ACCELERATION

http://www.mhhe.com/physsci/physical/giambatti
sta/circular/circular.html
The acceleration of an object in uniform circular
motion is:
It is important to note that the direction of the
change in velocity is always towards the center of the
circle.
Therefore the acceleration of an object in circular
motion is always towards the center of the circle. –
always!!!
This is the definition of centripetal, which means
center-seeking.
TANGENTIAL ACCELERATION
An object CAN have both tangential and
centripetal acceleration.
 If you drive a car around a curve at 45 km/hr,
there is centripetal acceleration.
 If you speed up (accelerate) to 50 km/hr, there is
centripetal AND tangential acceleration
 We will not consider tangential acceleration

Whenever an object is accelerated there must
be a net force acting on it.
This force is known as centripetal force, Fc.
This is not a new force, it is simply the net force
that accelerates an object towards the center
of its circular path.
Examples:
1. A mass is twirled in a circle at the end of a string, the
centripetal force is provided by…
tension
2. When a car rounds a corner on a highway, the
centripetal force is provided by…
friction
3. When the Moon circles the Earth, the centripetal force
is provided by…
gravity
On a FBD, label the centripetal force as specifically as you
can (not Fc)
Newton’s Second Law we can help us to determine a
formula for centripetal force:
Fc  m a c 
mv
r
2
m 4 r
2

T
2
Example:
A 0.50 kg mass sits on a frictionless table and is attached to
hanging weight. The 0.50 kg mass is whirled in a circle of
radius 0.20 m at 2.3 m/s. Calculate the centripetal force
acting on the mass.
Calculate the mass of the
hanging weight.
Example:
A car traveling at 14 m/s goes around an unbanked curve in
the road that has a radius of 96 m. What is its centripetal
acceleration?
What is the minimum coefficient of friction between the road
and the car’s tires in the last question?
EXAMPLE
FC=T=mv2/r
T 
( 0 . 90 kg )(19 m / s )
17 m
2
 19 N
19
A model airplane has a mass of 0.90 kg and
moves at a constant speed on a circle that is
parallel to the ground. Find the tension T in the
guideline(length=17m) for speed of 19 m/s.
HOW COULD YOU DETERMINE THE RADIUS OF A
CIRCLE IF ALL YOU HAD WAS AN OBJECT ON A
STRING, A SCALE, A STOPWATCH AND A FORCE
GAUGE (LIKE A SPRING SCALE)
Example:
A plane makes a complete circle with a radius of 3622 m in
2.10 min. What is the speed of the plane?
EXAMPLE. THE WALL EXERTS A 600 N FORCE ON
AN 80-KG PERSON MOVING AT 4 M/S ON A
CIRCULAR PLATFORM. WHAT IS THE RADIUS OF
THE CIRCULAR PATH?
Draw and label sketch
m = 80 kg;
v = 4 m/s2
Fc = 600 N
F 
r 
600 N
;
r
r
r=?
(80 kg)(4 m /s)
mv
2
2
r = 2.13 m
mv
F
2
THE CONICAL PENDULUM
A conical pendulum consists of a mass m
revolving in a horizontal circle of radius R
at the end of a cord of length L.
http://www.youtube.com/watch?v=5C4RJlFABic
T cos q
L
T
q
T
q
h
T sin q
R
mg
Note: The inward component of tension T sin q gives the needed
central force.
ANGLE Q AND VELOCITY V:
T cos q
L
T
q
T
q
h
T sin q
mg
R
Solve two
equations to find
angle q
T sin q 
T cos q = mg
mv2
R
v2
tan q = gR
EXAMPLE : A 2-KG MASS SWINGS IN A
HORIZONTAL CIRCLE AT THE END OF A CORD OF
LENGTH 10 M. WHAT IS THE CONSTANT SPEED OF
THE MASS IF THE ROPE MAKES AN ANGLE OF 300
WITH THE VERTICAL?
q
L
q
T
R
300
h
1. Draw & label sketch.
2. Recall formula for pendulum.
tan q 
v
2
gR
Find:
v=?
3. To use this formula, we need to find R = ?
R = L sin 300 = (10 m)(0.5)
R=5m
EXAMPLE 6(CONT.): FIND V
300
4. Use given info to find the
velocity at 300.
R=5m
g = 10 m/s2
Solve for v = ?
tan q 
FOR Q
q  300
L
T
v
=
q
R=5m
h
R
2
gR
v  gR tan q
2
v
2
g R tan q
v
(9.8 m /s )(5 m ) tan 30
0
v = 5.32 m/s
One last note on a little thing called centrifugal force.
While centripetal means center-seeking
centrifugal means center- fleeing.
Centrifugal force is actually an apparent force - it does not
exist. It is simply the apparent force that causes a rotating
object to move in a straight line.
However, Newton’s First Law tells us that we do not need a
force to keep an object moving in a straight line, you only
need a force to deflect an object from moving in a straight
line.
In reality what we seem to feel as centrifugal force is
really…
Example:
When riding in the backseat of a car that is turning a
corner, you slide across the seat, seeming to accelerate
outwards, away from the center of the turning circle.
In reality your forward inertia you had before the car
started to turn makes you want to continue in a straight
line (which makes you feel like you are sliding out)
When you slide into the side door, it exerts a centripetal
force (normal force in this case) and accelerates you
towards the center of the turn.