Circular Motion and
Gravitation
Holt Physics
Chapter 7
Centripetal Acceleration
• Circular motion:
motion of an object
that revolves about an
axis of rotation
Tangential Velocity
• Velocity (vt) of an
object directed along
a line tangent to its
circular path
• What would uniform
circular motion
mean?
vt depends upon radius
• Object B must travel
a greater distance to
keep up with object A
• SB > SA
• But ΔtB = ΔtA
• Therefore, vB > vA
Centripetal Acceleration
• Velocity is a vector (magnitude & direction)
• Acceleration is a change in velocity
• An object with constant speed, but
changing direction, is accelerating
• Acceleration of an object with uniform
circular motion (constant vt) has 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)
Centripetal Acceleration
• Centripetal
means “center
seeking” and is
always directed
toward the
center
• Due to a change
in direction of vt
2
vt
ac 
r
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
• Newton’s second law applies
• Fc=mac
• Since ac = vt2/r
• Fc=mvt2/r
• Centrifugal force is a misunderstanding of
inertia
Newton’s Universal Law of
Gravitation
• Gravitational Force
– Mutual force of attraction between particles of
matter
– Gravitational force is a very weak force
– is proportional to the masses of the objects
and inversely proportional to the square of
the distance between their centers.
Newton’s Universal Law of
Gravitation
m1m2
Fg  G 2
r
Fg is gravitational force
G is constant of universal gravitation;
G = 6.673 x 10-11N∙m2/kg2 (by Cavendish)
m1 m2 are masses of the two bodies
r is distance between centers of masses
Newton’s Universal Law of
Gravitation
Newton’s Mountain
Newton’s Thought
Experiment
A cannonball fired from
sufficient height with sufficient
velocity will orbit the earth.
Objects in orbit are in
continuous free fall
They do not hit the earth
because the earth curves
away from the object
•
•
http://spaceplace.nasa.gov/en/kids/orbits1.shtml
http://galileoandeinstein.physics.virginia.edu/more_stuff/Applets/newt/ne
wtmtn.html
Gravitational Field
• Gravity is a field force
• Gravitational Field
• Field strength varies
with distance from
Earth’s center
• Is described by g,
– i.e. the value of g
describes the strength of
the gravitational field at a
particular location in the
field
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 Second Law
The Law of Areas: A line that connects a planet to the sun sweeps
out equal areas in equal times
Kepler's Law Simulation
Kepler’s 3rd Law Describes Orbital
Period
Period and speed of an object in Circular O rbit
Orbital Period
Orbital speed
r3
m
T  2
vt  G
Gm
r
whe re m is the mass of the orbited mass
Weight & Weightlessness
• A bathroom scale reads the normal force
of the scale acting on your feet
• Under normal conditions this equals your
weight
• Suppose you stand on a scale…
• If someone were to push down on your
shoulders, how would the normal force
change?
• What if someone lifted you up somewhat?
• What if the scale was in an elevator?
• What happens when the elevator begins to
go up?
• What happens when it begins to go down?
• What would the scale read if the elevator
was in free fall?
•
•
•
•
In these situations, did your mass change?
Did the scale readings change?
So, did your true weight change?
What did change was your apparent
weight
Apparent Weight
Torque
• a quantity that measures the ability of a
force to rotate an object about an axis
• is not a force
• “turning 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
when 2 or more forces act to rotate the
same object, τnet = Στ
τnet = τ1 + τ2 = F1d1 + F2d2
Torque Equilibrium
• Sum of torques = 0
• ∑τ = 0; τ clockwise = τcounterclockwise
• An object is in “balance” like a teetertotter
Mobiles
• Are applications of torque equilibrium
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 = lever arm
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
Wout
eff 
Win
• A less efficient machine produces less output
per input
• A less efficient machine requires more input
to get the same output
Six Simple Machines
•
•
•
•
•
•
Lever
Inclined Plane
Wheel & axle
Pulley
Wedge
Screw