Circular Motion
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Transcript Circular Motion
• Why don’t riders of a roller
coaster fall out at the top of a
loop? … Does the speed of
the coaster have anything to
do with it?
• If a cup of water is swung
in a circle, why doesn’t the
water fall out? Does speed
matter here?
• When a car travels fast around a curve in a
road, why does a passenger get “thrown”
towards the far side of the car?
• If you do this in a golf cart with no door,
what will happen to the passenger?
Animations:
“Car” with
No Door
“Car” with
Door
In each of the previous examples, there
was a force present that caused the object
to travel in a circle.
• Without the force, the object in motion
continues along a straight-line path.
• With the force, the object in motion is
pulled or pushed towards the
___________________.
center of the circle This force is
then appropriately called a
centripetal force, which means
_____________
“center-seeking”.
Centripetal vs. Centrifugal Force
If an object traveling in a circle experiences a
force towards the center (centripetal), why then
do we feel like there is a force throwing us
away from the center when we travel in a
circular path?
We know that feeling is simply due to inertia
_______,
the tendency to continue in motion in a
straight line
_____________.
However, sometimes the term __________
centrifugal
center-fleeing is used
“force”, which means ______________,
A FORCE!!
to describe this, but it’s actually NOT
____________
Centripetal Acceleration
If an object traveling in a circle
experiences a centripetal force, then it
must experience a centripetal
acceleration as well.
___________
But… if an object is traveling in a circle at
constant speed, how can it be
accelerating?
Recall that acceleration is the rate
velocity changes.._______
direction
at which _______
is continually changing!
Centripetal Force Equation
Newton’s 2nd Law can then be used to find
an expression for the centripetal force that
causes the centripetal acceleration:
Fc
mac
mv
OR Fc
R
2
Linear Speed (or Velocity) Equation
How can we calculate the linear speed, v, of an
object traveling in a circle at a constant speed?
d
speed v
t
v
Where…
2R
2R
T
• _____ = circumference
Period the time for 1 revolution
• T = _______,
Recall that the direction of the velocity
vector is _______
tangent to the circular path.
Example Problem:
Linear Speed and Rotational Speed
Calculate the linear
speed, v, of a bug
riding on the edge of
an old 8 cm radius
record that is rotating
at a rotational speed
of 45 rpm.
1st find the period, T, in units of sec/rev:
rev 1 min
rev
sec
45
x
0.75
T reciprocal 1.3
min 60 s
sec
rev
2nd, find the speed, v:
2 (8 cm)
v
38 cm
s
1.3 s
• If the bug moved halfway in to R = 4 cm, find
the new linear speed. The rotational speed will
the same
be __________,
so the period, T, doesn’t
change.
2 (4 cm)
v
19 cm
s
1.3 s
•As the bug moves to the center, it’s linear
zero
speed will approach ______!
•To summarize, if the rotational speed (rpm’s) is
constant, as R ↑, linear speed, v ↑
__
If you
didn’t
follow
Calvin’s
dad the 1st
time, do
you now??
An Application: Train Wheels
In order to travel around a curve, a train’s outside
wheels must travel faster than it’s inside wheels,
(for the same reason that the starting points are
staggered in a race. If they weren’t staggered,
the runners on the outside would run a longer
distance!) But, if the train wheels are connected
to the same axle, how can they do that?
tapered
Train wheels are slightly _________,
and
the tracks are slightly rounded so that
only a small part of the wheel is actually
in contact with the track at any time!
A view of the tapered
wheels… looking
down the track
Picture from Paul Hewitt’s
Conceptual Physics
When a train makes a right turn, the tendency
of the train to go straight (Newton’s 1st Law of
Inertia!) “forces” the larger-diameter part of the
left wheel on the left track and the smallerdiameter part of the right wheel on the right
track. In 1 revolution of the axle, then, the left
wheel will travel a larger distance, and thus,
travels faster!
The Swings Ride at an Amusement Park
FBD:
Fc
T
W =mg
The centripetal force on a
horizontal
swing/rider is the _________
tension
component of the ________.
Why do the swings move farther
out as the speed goes up?
A car going around a flat curve
N
Ff
W
In order for a car to be able to go around a
flat curve, ________
friction must supply the
necessary centripetal force. If there is NOT
friction then the car will skid!
enough ________,
Equations:
2
Fc = F f = mv R AND
Ff =
N
IF NO SKID! (Static or Sliding?)
4. A car going around a banked curve
Fc
q
W
In addition to friction, we bank curves to make it
easier to “make” the curve.
horizontal component of the normal force
The __________
center of the circle, and thus
is towards the _______
significantly contributes to the centripetal force.
Satellite Physics – Circular Orbits
Isaac Newton was
the first to argue that
if a projectile is given
enough velocity, it
around
would fall _______
the earth rather than
into it!! The projectile
is then called a
________ and the
satellite
trajectory is called an
______.
orbit
Illustration from Isaac Newton,
Philosophiae Naturalis
Principia Mathematica, Book III
ONLY
So, what is the
force acting on a
satellite (planets, moons or man-made)?
gravitational force between the
The _____________
satellite and planet it’s orbiting (which
would equal the weight
______ of the satellite at
that point).
If the orbit is a circular orbit, then, the
gravitational force is the ___________
centripetal
force. (Note that the direction of the
gravitational force on the satellite is always
towards the center of the planet - it’s
“center-seeking”!)
But, wait…. If the force of gravity (or weight) is
the only force acting on a satellite, why do
astronauts experience what is known as
“weightlessness” (they appear to be floating!)
when in orbit?
NOT weightless Gravity is
Actually they are ______________!!
NOT
zero there (or they wouldn’t stay in
________
orbit!). They and the space shuttle are BOTH
falling
______. It would be more accurate to say:
They are experiencing
normalforcelessness or “__________
apparent
“____________________”
weightlessness”!
So, if gravity is the ONLY force
acting on the satellite, why doesn’t
the satellite “fall” and hit the earth?
IS falling! In
Remember that it ________....
order for a satellite to be moving
in a circular orbit, it must be
traveling at just the right velocity
______ to
fall around
the earth
___________
___________
We can calculate the required velocity, v,
for a satellite’s circular orbit:
V (NOT
W
a force)
Earth
R
R = radius of orbit (distance
from center of earth)
Fc 2 W
mv
mg
R
v gR
2
vsat gR
Mass of satellite __________
cancels
and thus has _______________
no influence
on the required speed! Why
does that make sense?
The “Rotor” Amusement Park Ride:
The “rotor” is spun in
a circle at a high
speed. At top
speed, the floor is
dropped, leaving the
riders “stuck” to the
wall! How does this
work??
What forces act on the rider and in what
direction do they act? Which FBD is
correct?
Ff,static
Normal
mg
So, the _____________
Normal Force is the centripetal
friction is the force that keeps
force, and ________
the person from sliding down.
Futuristic Space Stations:
One of the problems with people living
long-term in an orbiting space station is
the zero “apparent gravity” environment.
One potential solution to this problem is
to spin the orbiting space station about its
central axis in order to simulate gravity.
The centripetal force on a person in this
Normal force.
situation is a _______
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