Prof.
Clements Notes/Hints for Physics Courses
Chapter
6 Part 1 Open Stax College Physics or Most Introductory Physics
Courses
Rotation
Notation, Centripetal Acceleration and Force, Coriolis "Force",
Centrifugal "Force"
Terms
you should know at the end of this unit: arc, central angle, angular
velocity, centripetal, centrifugal, Coriolis
Angle
of Rotation and Angular Velocity
Think
about some objects that you have seen rotate: e.g. bicycle wheel,
Ferris Wheel, merry-go-round, etc. The spinning motion of these
objects leads to a different set of characteristics that describe the
rotation of the objects: angle of rotation, angular velocity, angular
acceleration, etc. The objects may also have a linear motion while
they are spinning (e.g. bicycle wheel) but I will generally only
discuss the rotation where the axis of rotation is fixed.
The
objects don’t travel in a straight line so you can’t use our 4
kinematic equations found in Chapter 2. But have no fear, you will
find that there exists 4 kinematic rotations for rotation that are
extremely similar to the familiar kinematic equations you have
mastered. And, the solution of problems for rotation will be
accomplished in a very similar manner with those equations.
Consider
the sector of a circle formed by the arc and the two radius lines
from the ends of the arc to the center of the circle.
Δθ
= Δ
S / r or S = r * θ
The angle formed between the two radius lines is the central angle,
θ. The central angle MUST be measured in radians for this
formula to apply. A reminder: 2π radians = 360 degrees. r is the
radius of the circle. S is the length of the arc opposite the
central angle. S and r must have the same units.
You
should know how to convert degrees to radians and radians to degrees
using the conversion factor 2π radians = 360 degrees
e.g.
45 degrees * 2π radians / 360 degrees = 0.785 radians
What
is the arc length if the radius is 1.6 meters and the central angle
is 45 degrees?
Answer:
S = rθ 45
degrees must be converted to radians, as shown above.
S = 1.6 meters * 0.785 radians = 1.26 meters
The
angular velocity is found by dividing the angle covered during the
motion by the time.
ω
= Δθ
/ Δt
Angular velocity has units of radians/second
As
the object moves around the circle it has a velocity tangent to the
circle. The magnitude of the velocity is related to the angular
velocity V = r * ω The units for ω must be radians/second.
e.g.
If ω is 3 radians/second and r is 1.2 meters then V is 3.6
meters/second.
Centripetal
Acceleration
You
should notice that the direction of the velocity changes as the
object moves in a circle. At any instant the velocity vector is
drawn "tangent" to the circular path of the object. The
velocity vector is perpendicular to the radius line that extends from
the object to the center of the circle. The magnitude of the velocity
is constant but the direction of the velocity changes. An
acceleration is needed to change velocity (speed or direction). The
name of the acceleration for uniform circular motion is centripetal
acceleration. Centripetal acceleration is directed towards the center
of the circular path of the object which is moving in circular
motion.
The
centripetal acceleration is calculated from ac = V2
/ r where r is the radius of the circle and V is the magnitude of
the velocity. Sometimes it is useful to use the rotation values to
calculate the centripetal acceleration. You can substitute V = r * ω
for V in the ac = V2 / r expression. When
you simplify you find that
ac
= r * ω2 . Take note that the acceleration is not
tangent to the circle.
What
happens to the value of the centripetal acceleration if the radius is
held constant but the velocity (m/s) increases by a factor of 3 (e.g.
from 8 m/s to 24 m/s)?
Answer:
Think about the equation: ac = V2 / r . ac
is proportional to the square of the velocity.
The
value of the original ac is ac = (1 * V)2
/ r or ac = 1 * V2 / r . If the velocity
increases by a factor of 3 the value of ac is now ac
= (3*V)2 / r or ac = 9 *V2 / r .
ac increases by 32 or 9 times the original
value for this example.
Centripetal
Force
As
you learned in Chapter 4, net external force causes acceleration.
The centripetal acceleration, ac, is caused by the
centripetal force, Fc. It is true that Fc = m
* ac
or Fc = mV2 / r . Note that
the centripetal force is perpendicular to the velocity.
Fc
only changes the direction of the velocity, it does not change the
magnitude (speed) of the velocity.
Examples
of centripetal force include
1.
Rubber stopper swung in a horizontal circle on the end of a string
The
string tension supplies the centripetal force.
2.
Car driving around a flat curve
Friction
of the road on the tires supplies the centripetal force.
3.
A satellite moving in a circular orbit around the Earth
The
force of gravity of the Earth on the satellite supplies the
centripetal force.
4.
Clothes in a washer during the spin cycle
The
wall of the rotating tub supplies an inward force on the clothes.
Why
does some water leave the clothes during the spin cycle?
And,
why does the tub of the washer have to spin so fast?
Answer:
The water molecules need an inward centripetal force to move in a
circle during the spin cycle. The adhesion force between the
water molecules and the clothes cannot provide enough centripetal
force when the velocity of spin is high. Recall: Fc = mV2
/ r . The water tends to move in a straight line instead of a
circle. The wall of the rotating tub in the washer constrains the
water somewhat but the water flows along the wall of the tub and out
through the holes on the wall of the tub. The clothes are too big
(we hope) to fit through the holes in the wall of the tub and don’t
leave the tub (except for an occasional sock !?).
The
tub has to spin fast to make the required value of Fc
larger than the available force from the adhesion of the water to the
clothes. The water will remain on the clothes and move in a circle
until the spin speed reaches a point where Fc = mV2
/r is larger than the adhesion force.
You
should work through the examples of centripetal acceleration for a
computer hard drive and the friction needed for a car on a curve
shown in the image below.
Fictitious
Forces and Non-inertial Frames: The Coriolis "Force"
If
you make observations in a non-inertial reference frame you will
think that fictitious forces exist. That is to say, you will observe
that objects don’t obey Newton’s laws of motion unless you invent
a force that is not really present. In a rotating system this
fictitious force is sometimes identified as an outward, centrifugal
force.
e.g.
The driver of a car making a sharp right turn will think there is a
force pushing the driver towards the door near the left arm.
Actually, your body is trying to move in a straight line and the car
is turning into your body.
e.g.
All parts of the earth make one rotation in 24 hours. The Earth
rotates towards the east (recall where the Sun rises!) . But,
objects located closer to the Earth’s equator are farther from the
rotation axis of the Earth that runs from the North Pole to the South
Pole through the center of the Earth. Objects closer to the equator
(as compared to objects in Nebraska, or Canada etc.) have a larger
value for r in the equation V = r * ω and thus have a larger
velocity in meters per second towards the east.
Imagine
you are at the Earth's equator. You are shooting a long range
artillery weapon that is aimed directly towards the north. Before
the weapon is fired the artillery shell and weapon are moving toward
the east due to the rotation of the Earth. The artillery shell will
keep its large eastward velocity due to the rotation of the Earth as
it is shot out the barrel of the weapon. As the shell moves towards
the north, it is traveling above land that has a smaller velocity
(meters/second) towards the east (recall V = r * ω ). The
observer on Earth will see the artillery shell land east of the
target because the shell is moving at a faster speed towards the east
than is the land it flies over. The observer will invent a
“Coriolis” force directed towards the east to explain why the
shell hit east of the target. The “Coriolis” "force"
is not a real force. The deviation of the shell toward the east is
just a consequence of the rotating Earth. This same effect is
responsible for the direction of rotation of high and low pressure
weather systems.
Centrifugal
"Force"
e.g.
Imagine rolling a ball across a merry-go-round in the playground
while you riding the merry-go-round. If friction is not a big effect
the ball will travel in a straight line as viewed by an observer on a
tree branch above the merry-go-round. But, the observer on the
merry-go-round will see the ball move in a curve such that the ball
moves towards the outer edge of the merry-go-round without passing
through the center. The observer will claim that there is an outward
centrifugal "force" that pushes the ball away from the
center. The centrifugal “force” does not exist.
YouTube
videos that explain the misconception of centrifugal "force."
Or
you may do your own search for videos that state that centrifugal
"force" is fictitious.
Here
are quantities that are uniform (constant) in uniform circular
motion: magnitude of velocity, magnitude of centripetal acceleration,
and magnitude of centripetal force are constant. The full vectors
for these items have a changing direction but have a constant length.
Other
constant items are radius and angular velocity. (The angular velocity
is a different sort of vector that will be discussed briefly in the
unit on angular momentum.)
What
force is necessary to keep the earth moving in a (nearly) circular
orbit around the sun?
Answer:
The force of gravity on the Earth due to the attraction to the Sun.
This force is directed towards the Sun.
e.g.
A Ferris wheel has a radius of 8 meters and rotates once in 15
seconds. What force does the seat of the Ferris wheel apply to a
passenger who has a mass of 80 kg and is currently at the highest
point in the ride?
Copyright©
2017 by Greg Clements Permission is granted to reproduce this
document as long as 1) this copyright notice is included, 2) no
charge of any kind is made, and, 3) the use is for an educational
purpose. Editing of the document to suit your own class style and
purposes is allowed.
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