Objectives: The student will be able to:
• identify uniform circular motion.
• determine the directions of the velocity and
acceleration vectors for an object in uniform
circular motion.
• calculate the centripetal acceleration of a point
mass in uniform circular motion given the radius of
the circle and either the linear speed or the period
of the motion.
• state the relationship between the period of the
motion and the frequency of rotation and express
this relationship using a mathematical equation.
Uniform Circular Motion: Definition
Uniform circular motion
Constant speed, or,
constant magnitude of velocity
March 22, 2016
Motion along a circle:
Changing direction of velocity
Uniform Circular Motion
Uniform circular motion is motion along a
circular path in which there is no change in
speed, only a change in direction.
Fc
v
Constant velocity
tangent to path.
Constant force
toward center.
Question: Is there an outward force on the ball?
Uniform Circular Motion (Cont.)
The question of an outward force can be
resolved by asking what happens when the
string breaks!
Ball moves tangent to
v
path, NOT outward as
might be expected.
When central force is removed,
ball continues in straight line.
Centripetal force is needed to change direction.
Kinematics of Uniform Circular Motion
Uniform circular motion: motion in a circle of
constant radius at constant speed
Instantaneous velocity is always tangent to
circle.
Kinematics of Uniform Circular Motion
This acceleration is called the centripetal, or
radial, acceleration, and it points towards the
center of the circle.
Examples of Centripetal Force
You are sitting on the seat next to
the outside door. What is the
direction of the resultant force on
you as you turn? Is it away from
center or toward center of the turn?
• Car going around a
curve.
Fc
Force ON you is toward the center.
Car Example Continued
Reaction
Fc
F’
The centripetal
force is exerted
BY the door ON
you. (Centrally)
There is an outward force, but it does not act
ON you. It is the reaction force exerted BY you
ON the door. It affects only the door.
Another Example
Spin Cycle on a Washer
How is the water removed
from clothes during the
spin cycle of a washer?
Think carefully before answering . . . Does the
centripetal force throw water off the clothes?
NO. Actually, it is the LACK of a force that
allows the water to leave the clothes
through holes in the circular wall of the
rotating washer.
Example 1: A 3-kg rock swings in a circle
of radius 5 m. If its constant speed is 8
m/s, what is the centripetal acceleration?
2
v
v
m
m = 3 kg
ac =
r
r
r = 5 m; v = 8 m/s
2
(8 m/s)
2
ac 
 12.8 m/s
5m
mv
Fc = mac =
r
2
F = (3 kg)(12.8 m/s2)
Fc = 38.4 N
Example: The Effect of Radius on
Centripetal Acceleration
The bobsled track at the 1994
Olympics in Lillehammer,
Norway, contained turns with
radii of 33 m and 24 m, as the
figure illustrates. Find the
centripetal acceleration at each
turn for a speed of 34 m/s, a
speed that was achieved in the
two-man event. Express the
answers as multiples of
g=9.8m/s2.
12
From ac=v2/r it follows that
Radius=33
m
2
(34m / s)
ac 
 35m / s 2  3.6 g
33m
Radius=24
m
2
(34m / s)
ac 
 48m / s 2  4.9 g
24m
13
Check your understanding 1
The car in the drawing is moving clockwise around a
circular section of road at a constant speed. What are the
directions of its velocity and acceleration at (a) position 1
and (b) position 2?
14
(a) The velocity is due south, and the
acceleration is due west.
(b) The velocity is due west, and the
acceleration is due north.
15
Uniform Circular Motion
Transparency
16
Period and Frequency
A circular motion is described in terms of the
period T, which is the time for an object to
complete one revolution.
2r
T
v
v= 2r/T = 2rf
The distance traveled in one revolution is
2 r
r
The frequency, f, counts the number of revolutions
per unit time.
1
f
T
Tangential (Linear) Velocity
The tangential velocity vector is
tangent to the circle at the point of
v
study.
v
Problem 1
A biker travels once around a circular
track of radius 20.0m in 3s.
Calculate:
a)
the average tangential speed
b)
the frequency
c)
the period
Answers: 41.9m/s, f=0.33Hz, T=3s
Record Player
Problem 2
A coin sits 0.10m from the center of a
record player spinning at 45rpm.
a)
What is the frequency in Hertz?
b)
What is the period?
c)
What is the linear speed?
Answer: 0.75Hz, 1.33s, 0.47m/s
Uniform Circular Motion: Observations

Object moving along a curved
path with constant speed
 Magnitude of velocity: same
 Direction of velocity:
changing

 Velocity v : changing
 Acceleration is NOT zero!
 Net force acting on an
object is NOT zero

“Centripetal force”
March 22, 2016


Fnet  ma
Dynamics of Uniform Circular Motion
For an object to be in uniform circular motion,
there must be a net force acting on it.
We already know the
acceleration, so can
immediately write the
force:
Uniform Circular Motion
Newton’s 2nd Law: The net force on a body is
equal to the product of the mass of the body and
the acceleration of the body.
*The centripetal acceleration is
caused by a centripetal force
that is directed towards the
center of the circle.
F  ma  m
2
v
r
Problem 3
A child on a merry-go-round sits 1.5m
from the center. They spin 3 times in one
min. The mass of the child is 40kg.
Find the friction(centripetal force) acting on
the child.
Answer: 5.9N
Dynamics of Uniform Circular Motion
We can see that the force must be inward by
thinking about a ball on a string:
Demo – penny
and hanger
Dynamics of Uniform Circular Motion
There is no centrifugal force pointing outward;
what happens is that the natural tendency of the
object to move in a straight line must be
overcome.
If the centripetal force vanishes, the object flies
off tangent to the circle.
Period 3 start here.
“Centrifugal Force”
• “centrifugal force” is a fictitious force it is not an interaction between 2
objects, and therefore not a real force.
• Nothing pulls an object away from
the center of the circle.
“Centrifugal Force”
• What is erroneously attributed to
“centrifugal force” is actually the action
of the object’s inertia - whatever
velocity it has (speed + direction) it
wants to keep.
What provides Centripetal Force ?
• Centripetal force is not a new kind of force
• Centripetal force refers to any force that keeps
an object following a circular path
mv 2
Fc  mac 
r
• Centripetal force is a combination of
– Gravitational force mg: downward to the ground
– Normal force N: perpendicular to the surface
 and away from
– Tension force T: along the cord
object
– Static friction force: fsmax = µsN
March 22, 2016
What provides Centripetal Force ?
Fnet  N  mg  ma
2
v
N  mg  m
r
Fnet  T  ma 
mv
T
r
March 22, 2016
N
a
v
2
mg
Circular Motion Lab
Do not
use
clamp.
Highway Curves, Banked and Unbanked
When a car goes around a curve, there must be
a net force towards the center of the circle of
which the curve is an arc. If the road is flat, that
force is supplied by friction.
Highway Curves, Banked and Unbanked
If the frictional force is
insufficient, the car will
tend to move more
nearly in a straight line,
as the skid marks show.
Highway Curves, Banked and Unbanked
As long as the tires do not slip, the friction is
static. If the tires do start to slip, the friction is
kinetic, which is bad in two ways:
1. The kinetic frictional force is smaller than the
static.
2. The static frictional force can point towards
the center of the circle, but the kinetic frictional
force opposes the direction of motion, making
it very difficult to regain control of the car and
continue around the curve.
Car Negotiating a Flat Turn
v
Fc
R
What is the direction of the
force ON the car?
Ans. Toward Center
This central force is exerted
BY the road ON the car.
Car Negotiating a Flat Turn
v
Fc
R
Is there also an outward force
acting ON the car?
Ans. No, but the car does exert a
outward reaction force ON the road.
Car Negotiating a Flat Turn
The centripetal force Fc is
that of static friction fs:
m
Fc
R
n
fs
Fc = fs
R
v
mg
The central force FC and the friction force fs
are not two different forces that are equal.
There is just one force on the car. The nature
of this central force is static friction.
Finding the maximum speed for
negotiating a turn without slipping.
n
fs
Fc = fs
m
v
R
Fc
R
mg
The car is on the verge of slipping when FC is
equal to the maximum force of static friction fs.
Fc = fs
Fc =
mv2
R
fs = msmg
Optimum Banking Angle
By banking a curve at the
optimum angle, the normal
Fc
m
R
v
fs
w
force n can provide the
necessary centripetal force
without the need for a
friction force.
n
q
slow speed
n
w
fs
q
fast speed
fs = 0
w
n
q
optimum
Highway Curves, Banked and Unbanked
Banking the curve can help keep
cars from skidding. In fact, for
every banked curve, there is one
speed where the entire centripetal
force is supplied by the
horizontal component of
the normal force, and no
friction is required. This
occurs when:
Class work / Homework
 Hand-outs
 Page 156 12-21
Closure
 Kahoot Circular Motion