http://www.physicsclassroom.com/mmedia/circmot/circmotTOC.html
Circular Motion
Your body cannot sense constant velocity.
(Earth, windowless car) But, your body is
very good at feeling acceleration(punch
gas pedal or brake)
Acceleration is change in velocity. Up to
now, that has been on one direction. We
will see that one can have constant speed
and acceleration at the same time.
A constant speed and changing direction
is also acceleration.
Period…
Period is the time needed for
one complete cycle.
T is the symbol for period
…
When something is in a curve, it changes its
direction every instant.
When any object is in a curve or revolving,we say it
is acceleration (even if speed remains the same)
We call this acceleration CENTRIPETAL
ACCELERATION Ac
Centripetal Force
Centripetal Force is the force that
keeps an object moving in a circular
path. Remove the force and the
object will NOT move in a circle.
Ball & String
Car & Road
Earth & Gravity
When Fc is removed, the net force
has been removed and the object
travels in a straight line
Centripetal is Center Seeking
“Trip to center”
When you observe objects
moving away from the center
of curvature, they do so b/c
they do NOT have enough Fc
a = ΔV/ Δt
ac = ΔV/ Δt
ac = v2/ r
Velocity for something turning
V = d/t = circumference/ period
V = 2πr/ T
Substitute V into our ac equation
ac = (2πr/ T)2
ac = 4π2r / T2
r
Newton’s 2nd law says f= ma
Therefore centripetal force = mac
Fc = mac
Fc = mV2/ r
Fc = m(4π2r / T2)
Example Problem p.49
A 0.013kg rubber stopper is
attached to a 0.93m string. The
stopper is swung in a horizontal
circle, making 1 revolution in 1.18s.
a. Find the speed for the stopper
b. Find the stopper’s centripetal
acceleration
c. Find the force the string exerts on the
stopper
Given
m = 0.013kg
Formula
r = 0.93m
V = 2πr/ T
T = 1.18sec
ac = V2/r
V=?
ac = ?
Fc = ?
Fc = Mac
Solution
a. 2(3.14)(.93m)/ 1.18 = 5m/s
b. (5m/s)2/ .93 = 26.9m/s2
c. Fc = (.013)(26.9) = .35N
inward
Class Work
1. Sue is swinging a yo-yo around her head.
What happens to the size of the ac if the
mass of the yo-yo is doubled without
changing the period or length of string?
2. Imagine you are in a car seat tossing a
ball straight up in the air.
If the car is moving with constant velocity, will
the ball land in front, behind or in your hand?
If the car is decelerating in a straight line , will
the ball land in front, behind, or in your hand?
If the car rounds a curve at constant speed,
where will the ball land?
3. Geraldo whirls a 20g rubber stopper above
his head. The string that holds the stopper is
1.5m long. The rate of spin is 3 revolutions
every second.
Find ac of the stopper
Find linear velocity
Find tension in string (Fc)
4. A 100kg pilot is in an airplane doing a loop de
loop. The jet has a velocity of 88m/s. The
radius of the loop is 200m
What is the pilot’s weight?
what is the pilots ac?
What is the pilots Fc?
compare the pilots Fc to his weight
Class work Solutions
1. Nothing, ac is independent of mass
2. A= In your Hand
B= In front of your hand
C= To the side of your hand. The
outside of the curve
Given
Formula
m = .02kg
(A) ac = 4π2r/
T2
r= 1.5m
(B)V = 2πr/ T
3 revs/sec
T= 1/3 =
.33sec
(C) Fc = mac
Solution (A)
Solution (B)
Solution (C)
39.44(1.5m)
(.33)2
2(3.14)1.5
.02 (543.3)
.33
=10.9N
ac = 543. 3m/s2
V = 28.5m/s
Given
Formula
m = 100kg
W = mg
V = 88m/s
ac = V2/r
r = 200m
Fc = Mac
Solution (A)
Solution (B)
Solution(C)
100(9.8)
(88)2/ 200
100(38.7)
= 980 N
ac = 38.7m/s2
=3872N
compare: 3872/ 980 = 3.95
Fc is 3.95x greater – aka a 4g turn
??
An early objection to the idea that
Earth is spinning on its axis was that
Earth would turn so fast at the
equator that people would be thrown
off into space. Show the error in this
logic by calculating…
The speed of a 97kg person at the
equator, the radius of Earth is 6400Km
The centripetal Force on the person
The weight of the person
Given
Formula
r = 6,400,000m
(A) V = 2πr/T
M = 97kg
(B) Fc = m(v2/r)
(C) W = mg
Solution (A)
2(3.14)(6,400,000)
24hr(3600s)
V= 465 m/s
Solution (B)
Fc = 97(4652/ 6,400,000)
Fc = 3.28N
Solution (C)
97kg(9.8m/s2)
= 950.6 N
3.3/ 950.6 =
Fc is only .3% of the
person’s weight
-too small to worry about
Simple Harmonic Motion
A playground swing, vibrating
spring, guitar string & pendulum,
are all examples of simple harmonic
motion.
SHM occurs if the restoring force
varies linearly with the
displacement.
Uniform Circular Motion: Period
Object repeatedly
finds itself back
where it started.
distance = rate  time
distance 2r
time =

rate
v
2r
T=
v
The time it takes to
travel one “cycle” is
the “period”.
Tension Can Yield a Centripetal Acceleration:
If the person doubles the
speed of the airplane,
what happens to the
tension in the cable?
mv
F = ma 
r
Doubling the speed, quadruples the
force (i.e. tension) required to keep
the plane in uniform circular motion.
2
Centripetal Force: Question
A car travels at a constant
speed around two curves.
Where is the car most likely to
skid? Why?
mv
F = ma 
r
2
Smaller radius: larger force
required to keep it in uniform
circular motion.
Q: Why exit ramps in highways are banked?
Vertical Circular Motion