Transcript Circular Motion & Gravity

Circular Motion & Gravity
Circular Motion
• Objects travel in a circle
• Rotate about an axis of rotation
• Tangential speed (vt) describes the rate at
which the object moves around the circle
2r circumfere nce
vt 

T
period
• Direction is tangential to the circular path
vt depends upon radius
• Given the object is
rigid, e.g. a CD
• Object B must travel
a greater distance to
keep up with object A
• SB > SA
• But ΔtB = ΔtA
• Therefore, vB > vA
Comparison of Translational Motion
& Uniform Circular Motion
UCM = motion of an object traveling in a circle at a
constant speed, vt
Type of Motion
Translational
Displacement
Linear
Δx
Time
Δt
Formula
vavg = Δx/Δt
Uniform
Circular
Circumference
2πr
Period
T
vt = 2πr/T
Uniform Circular Motion
• Tangential speed vt
is constant
• Because direction
is changing, there
is acceleration
• Centripetal
acceleration
Centripetal Acceleration
• a = Δv/Δt
• When subtracting
vectors, reverse the
direction of vi
• Centripetal
acceleration is,
therefore, directed
toward the center
(axis of rotation) when
θ is small
Centripetal Acceleration
• Centripetal means
“center seeking” and is
always directed toward
the center
• Due to a change in
direction of vt
• Phet simulation
2
vt
ac 
r
Tangential Acceleration
• Tangential acceleration
occurs when there is a
change in tangential
speed.
• For example, if a car is
speeding up as it goes
around a curve,
– It has tangential
acceleration and
– Centripetal acceleration
Centripetal Force
Because Fc acts at right angles
to the object’s circular motion, it
changes the direction of the
objects velocity
Centripetal Force
• Is the cause of centripetal acceleration
• It is directed toward the axis of rotation
• It is the net force acting on an object in
uniform circular motion, i.e. it is the cause
of circular motion
• Centrifugal force is a misunderstanding
of inertia
Centripetal Force
& Newton’s 2nd Law
F  ma
2
vt
Since ac 
r
Fc  mac
2
vt
Fc  m
r
Centripetal Force
• Is just the name of any net force acting on
an object in uniform circular motion
• Fc could take any form….
• It could be frictional force, tension force,
gravitational force, etc.
Motion of a Car Around a
Curve
• On a horizontal turn, the centripetal force is
friction
Circular Motion About a Banked
Curve
Conical Pendulum
Vertical Circular Motion
Centifugal Force?
• If Fc is insufficient to maintain circular
motion, the object will leave it’s circular
path due to its own inertia, not because
some force is pulling it away from the axis
of rotation
• Thus, inertia is often mistaken for
“centrifugal force”
Gravity
Gravitational Force
•
•
•
•
•
Force of attraction between two masses
Attractive only
One of four fundamental forces
Very weak (the weakest)
When one object orbits another,
gravitational force is a centripetal force
Newton’s Law of Universal
Gravitation
• Gravitational force is…
– directly proportional to the product of the masses of
the two bodies
– inversely proportional to the square of the distance
between the centers of the two masses
– If the objects are large (e.g. planets, moons) then the
radii would be included in r
m1m2
Fg  G 2
r
G  universal gravitatio n constant
2
N

m
G  6.673 10 11
kg 2
Gravitational Force Exists Between
Any Two Masses
Newton’s Cannon
http://spaceplace.nasa.gov/en/kids/orbits1.shtml
http://galileoandeinstein.physics.virginia.edu/more_stuff/Applets/n
ewt/newtmtn.html
Importance of Gravitational Force
• Keeps you from floating away into space
• Gravitational force keeps the Moon and
planets in orbit
• Keeps earth in orbit around sun
• Causes ocean tides
Black Holes: Extreme Gravity
Extreme density
Escape velocity >
speed of light
Detect by effects
on surrounding
matter
Gravitational Field Strength
• Increases as distance
from mass center
decreases
• Because gravitational
field strength varies,
weight varies with
location
Gravitational Field Strength
• Describes the amount of gravitational
force per unit mass at any given point
• Equals free-fall acceleration
g
Fg
m
Weight Changes with Location
• Because gravitational field strength varies, ag
varies (acceleration of gravity).
• Since w = mag, weight must vary as ag varies
• Fg is an example of an inverse square law
m1m2
Fg  G 2
r
7.3 Motion in Space
Astronomer
Ptolomey
Planets
orbit…
Earth
Type of
orbit
Epicycles
Copernicus
Sun
Circular
Kepler
Sun
Elliptical
Kepler’s Laws of Planetary Motion
1. The Law of Orbits: All planets move in
elliptical orbits, with the sun at one focus.
2. The Law of Areas: A line that connects a
planet to the sun sweeps out equal areas in
equal times.
3. The Law of Periods: The square of the
period of any planet is proportional to the
cube of the average distance from the sun,
T 2  r3
Kepler’s
st
1
Kepler's Law Simulation
and
nd
2
Laws
Kepler’s 3rd Law Describes Orbital
Period
Period and speed of an object in Circular O rbit
Orbital Period
3
Orbital speed
r
m
T  2
vt  G
Gm
r
whe re m is the mass of the orbited mass
Actual and Apparent Weight
• A bathroom scale records the normal force
of scale acting on your body
• Step on the scale … the normal force
equals your weight
Actual and Apparent Weight
• Now try this
• Step on the scale and have someone
press down on your shoulders
– Predict and explain the result
• Step on the scale and have someone lift
you slightly
• Predict and explain the result
Actual and Apparent Weight
• How does this relate to your experiences
in an elevator?
• What would the scale read if, in an
elevator, it descended with an acceleration
of g?
Weight and Apparent Weightlessness
Torque
• a quantity that measures the ability of a
force to rotate an object about an axis
• is not a force
• “rotating ability”
• the product of force and “lever arm”
• τ = F · d sinθ
• Lever arm (d) is distance perpendicular to
direction of force to axis of rotation
  Fd sin 
Torque
• Sign
(+) is counterclockwise
(-) is clockwise
• Net Torque and
when 2 or more forces act to rotate the
same object, τnet = Στ
τnet = τ1 + τ2 = F1d1 + F2d2
Torque Equilibrium
• Torque Equilibrium: Στ = 0
Torque Equilibrium
The torque due to the boy is equal and opposite to that
of the girl.
Net Torque
Center of Mass (COM)
• Point mass vs. extended object
• The point in a body at which all the mass
can be considered to be concentrated
when analyzing translational motion
• Unless an object rotates about a fixed
point, (e.g. a hinge)…
– The point about which a mass or system of
mass rotates during rotational motion
Center of Mass
• The extended object rotates about the
CoM
• CoM follows the expected parabolic path
Center of Mass
• May not lie within the mass or system of
masses
Simple Machines
• All machines are combinations of simple machines
• Purpose is to change magnitude or direction of an
input force
• Mechanical Advantage
describes the ratio of output and input forces
Fout
MA 
Fin
Ideal vs. Actual Mechanical
Advantage
• Ideal MA
MA if there were no friction
d in
IMA 
d out
• Actual MA
MA that takes friction into
account
Fout
AMA 
Fin
Machines and Work
• Machines do not change the amount of work
• Machines make work easier
Efficiency
• A measure of how well a machine works
• A less efficient machine produces less output
per input
• A less efficient machine requires more input
to get the same output
Wout
eff 
Win