Transcript Circular Motion - Effingham County Schools

Introduction to
Uniform Circular Motion
Uniform Circular Motion
An object moves at uniform
speed in a circle of constant
radius.
Uniform circular motion is
accelerated motion. Why?
Checking Understanding
When a ball on the end of a string is swung in a vertical
circle, the ball is accelerating because
A.
B.
C.
D.
the
the
the
the
speed is changing.
direction is changing.
speed and the direction are changing.
ball is not accelerating.
Slide 6-13
Answer
When a ball on the end of a string is swung in a vertical
circle, the ball is accelerating because
A.
B.
C.
D.
the speed is changing.
the direction is changing.
the speed and the direction are changing.
the ball is not accelerating.
Slide 6-14
Centrifugal Force
It’s a myth!
We need to go back to
Newton’s Laws to properly
explain the feeling you get on
a merry-go-round or in a
turning car.
When a car accelerates
You, as a passenger,
feel as if you are flung
backward.
Your inertia (mass)
resists acceleration.
You are NOT flung
backward. Your body
wants to remain at rest
as the car accelerates
forward.
When a car decelerates
You, as a passenger, feel as
if you are flung forward.
Your inertia (mass) resists
the negative acceleration.
You are NOT flung
forward. Your body wants
to remain in motion at
constant velocity as the
car accelerates backwards.
When a car turns
You feel as if you are flung to the
outside. Your inertia resists acceleration.
You are not flung out, your body simply
wants to keep moving in straight line
motion!
As a general rule
Whenever you feel you are flung in a
certain direction, you can bet the
acceleration is pointing in the
opposite direction.
Remember the elevator problems?
When you feel you are flying up,
acceleration of the elevator is down.
When you feel you are sinking down,
acceleration is up.
Acceleration in Uniform
Circular Motion
The velocity vector at any given
point is subjected to an acceleration
that turns it, but does not speed it
up or slow it down.
The acceleration vector is always at
right angles to the velocity.
The acceleration points toward the
center of the circle.
Acceleration in Uniform
Circular Motion
The acceleration responsible
for uniform circular motion
is referred to as centripetal
acceleration.
Checking Understanding
When a ball on the end of a string is swung in a vertical
circle:
What is the direction of the acceleration of the ball?
A.
B.
Tangent to the circle, in the direction of the ball’s
motion
Toward the center of the circle
Slide 6-15
Answer
When a ball on the end of a string is swung in a vertical
circle:
What is the direction of the acceleration of the ball?
A.
Tangent to the circle, in the direction of the ball’s
motion
B. Toward the center of the circle
Slide 6-16
• ac =
Centripetal
Acceleration
2
v /r
v
ac: centripetal
acceleration in m/s2
v ac
v: tangential speed in m/s
r: radius in metersv
V = 2πr / T
T=period
a
c
ac
v
Centripetal acceleration always points toward center of
circle!
Force in Uniform Circular
Motion
A force responsible for uniform
circular motion is referred to as
a centripetal force.
Centripetal force is simply mass
times centripetal acceleration.
Fc = mac
Centripetal Force
• Fc = m ac
2
• Fc = m v / r
Fc: centripetal
force in N
v: tangential speed
in m/s
r: radius in meters
v
Fc
v
Fc
Fc
v
Always toward
center of circle!
More on Centripetal
Force
Centripetal force is not a unique
type of force.
Centripetal forces always arise from
other forces.
You can always identify the real force
which is causing the centripetal
acceleration.
Nearly any kind of force can act as a
centripetal force.
Friction as centripetal force
As a car makes a
turn, the force of
friction acting upon
the turned wheels of
the car provide the
centripetal force
required for circular
motion.
Example Problem
A level curve on a
country road has a
radius of 150 m. What
is the maximum speed
at which this curve
can be safely
negotiated on a rainy
day when the
coefficient of friction
between the tires on a
car and the road is
0.40?
Slide 6-28
Tension as centripetal force
As a bucket of
water is tied to a
string and spun in a
circle, the force of
tension acting upon
the bucket provides
the centripetal
force required for
circular motion.
Gravity as centripetal force
As the moon orbits the
Earth, the force of
gravity acting upon the
moon provides the
centripetal force
required for circular
motion.
Normal force as centripetal
force
An automobile turning
on a banked curve uses
the normal force to
provide the necessary
centripetal force.
Example Problem
A curve on a racetrack of radius 70 m
is banked at a 15 degree angle. At
what speed can a car take this curve
without assistance from friction?
Weight on a string
moving in vertical
circle
Centripetal
force arises
from
combination
of tension
and gravity.
Tennessee Tornado at
Dollywood
Centripetal force
when you are upside
down arises from a
combination of
normal force and
gravity.
Centripetal Force can do no
work
A centripetal force alters the
direction of the object without
altering its speed.
Since speed remains constant,
kinetic energy remains constant,
and work is zero.
The Universal Law of Gravity
Fg = Gm1m2/r2
 Fg:
Force due to gravity (N)
 G: Universal gravitational
constant
6.67
x 10-11N m2/kg2
 m1 and m2:
the two masses (kg)
 r: the distance between the
centers of the masses (m)
The Force of Gravity
Slide 6-35
Acceleration due to gravity
2
/r
Fg = mg = GmME
What is g equivalent
to?
g
2
= GME/r
Example Problems
1) A typical bowling ball is spherical, weighs 16 kgs, and has a
diameter of 8.5 m. Suppose two bowling balls are right next
to each other in the rack. What is the gravitational force
between the two—magnitude and direction?
2) What is the magnitude and direction of the force of
gravity on a 60 kg person? (Mearth = 5.98x1024 kg, Rearth =
6.37 x 106 m)
Slide 6-36
Acceleration and distance
Planet
Radius(m Mass (kg) g (m/s2)
Mercury
2.43 x 106 3.2 x 1023
Venus
6.073 x
4.88 x1024 8.83
106
3.38 x 106 6.42 x 1023 3.75
Mars
Jupiter
3.61
Saturn
6.98 x 106 1.901 x
26.0
1027
5.82 x 107 5.68 x 1026 11.2
Uranus
2.35 x 107 8.68 x 1025 10.5
Neptune
2.27 x 107 1.03 x 1026 13.3
Pluto
1.15 x 106 1.2 x 1022
0.61
Kepler’s Laws
1.
2.
3.
Planets orbit the sun in elliptical
orbits.
Planets orbiting the sun carve out
equal area triangles in equal times.
The planet’s year is related to its
distance from the sun in a
predictable way.
Kepler’s Laws
Satellites
Orbital Motion
•
•
Gmems / r2 = mev2 / r =
The mass of the orbiting body
does not affect the orbital
motion!
Consider the see saw
Consider the see saw
Consider the see saw
Consider the see saw
The weight of each child is
a downward force that
causes the see saw to twist.
The force is more effective
at causing the twist if it is
greater OR if it is further
from the point of rotation.
Consider the see saw
The twisting force,
coupled with the distance
from the point of rotation
is called a torque.
What is Torque?
Torque is a “twist” (whereas
force is a push or pull).
Torque is called “moment” by
engineers.
The larger the torque, the
more easily it causes a
system to twist.
Torque
Consider a beam connected to a wall by a hinge.
Now consider a
force F on the
beam that is
applied a distance r
from the hinge.
Hinge (rotates)
r
Direction of
rotation
F
What happens? A rotation occurs due to the
combination of r and F. In this case, the
direction is clockwise.
Torque
If we know the angle the force acts at, we
can calculate torque!
Hinge (rotates)
r
 = F r sin 
Direction of
  is torque
rotation
 is force
 r is “moment arm”
  is angle between F and r

F
Torque equation: simplified
Hinge: rotates
If  is 90o…
r
=Fr
Direction of
  is torque
rotation
 F is force
 r is “moment arm
 F
We use torque every day
Consider the door to the classroom.
We use torque to open it.
Identify the following:
 The point of rotation.
 The moment arm.
 The point of application of force.
Question
Why is the doorknob far from the
hinges of the door? Why is it not in
the middle of the door? Or near
the hinges?
Torque Units
What are the SI units for
torque?
 mN
or Nm.
Can you substitute Joule for Nm?
 No.
Even though a Joule is a Nm, it
is a scalar. Torque is a vector and
cannot be ascribed energy units.
Now consider a
balanced situation
40 kg
40 kg
If the weights are equal, and the moment arms are
equal, then the clockwise and counterclockwise
torques are equal and no net rotation will occur. The
kids can balance!
Now consider a
balanced situation
40 kg
40 kg
ccw = cw
This is called rotational equilibrium!
Periodic Motion
•
•
Repeats itself over a fixed
and reproducible period of
time.
Mechanical devices that do
this are known as
oscillators.
An example of periodic
motion…
A weight attached to a spring
which has been stretched and
released.
• If you were to plot distance
the vs time you would get a
graph that resembled a sine
or cosine function.
•
An example of periodic
motion…
3
2
-3
x(m)
4
6
t(s)
Simple Harmonic Motion
(SHM)
Periodic motion which can be
described by a sine or cosine
function.
• Springs and pendulums are
common examples of Simple
Harmonic Oscillators (SHOs).
•
Equilibrium
•
•
The midpoint of the
oscillation of a simple
harmonic oscillator.
Position of minimum
potential energy and
maximum kinetic energy.
All oscillators obey…
Law of
Conservation of
Energy
Amplitude (A)
•
•
How far the wave is from
equilibrium at its maximum
displacement.
Waves with high amplitude
have more energy than waves
with low amplitude.
Period (T)
•
The length of time it takes for
one cycle of periodic motion to
complete itself.
Frequency (f):
How fast the oscillation is
occurring.
• Frequency is inversely related to
period.
• f = 1/T
• The units of frequency is the
Herz (Hz) where 1 Hz = 1 s-1.
•
Parts of a Wave
T
3
A
-3
2
Equilibrium point
x(m)
4
6
t(s)
Springs
A very common type of Simple
Harmonic Oscillator.
Our springs will be ideal springs.
They are massless.
 They are compressible and extensible.

They will follow a law known as
Hooke’s Law.
Restoring force
The restoring force is the secret
behind simple harmonic motion.
The force is always directed so
as to push or pull the system
back to its equilibrium (normal
rest) position.
Hooke’s Law
A restoring force directly
proportional to displacement is
responsible for the motion of a
spring.
F = -kx
where
F: restoring force
k: force constant
x: displacement from equilibrium
Hooke’s Law
Fs = -kx
m
Fs
mg
The force constant of a
spring can be determined
by attaching a weight and
seeing how far it
stretches.
Hooke’s Law
Equilibrium position
F
m
x
F = -kx
Spring compressed,
restoring force out
Spring at equilibrium,
restoring force zero
m
F
x
m
Spring stretched,
restoring force in
Period of a spring
T = 2m/k
T:
period (s)
m: mass (kg)
k: force constant (N/m)
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Pendulums
The pendulum can be thought of
as an oscillator.
The displacement needs to be
small for it to work properly.
Pendulum Forces

T
mg sin
 mg
Period of a pendulum
T = 2l/g
T:
period (s)
l: length of string (m)
g: gravitational acceleration
(m/s2)