Transcript Circular Motion

Rotational Motion and
The Law of Gravity
Ch 7
Rotation and Revolution
• Two types of circular motion are rotation
and revolution.
• An axis is the straight line around which
rotation takes place.
• When an object turns about an internal
axis—that is, an axis located within the
body of the object—the motion is called
rotation, or spin.
• When an object turns about an external
axis, the motion is called revolution.
• The Ferris wheel turns about
an axis.
• The Ferris wheel rotates,
while the riders revolve about
its axis.
• Earth undergoes both types
of rotational motion.
• It revolves around the sun
once every 365 ¼ days.
• It rotates around an axis
passing through its
geographical poles once
every 24 hours.
Rotational Motion
• Solid objects undergo rotational motion
• A point on a rotating object undergoes
circular motion
• Circular Motion is described in terms of the
angle through which the point on the
object moves
Centripetal Acceleration
• Tangential speed (vt)
depends on distance
• When tangential
speed is constant,
motion is described
as uniform circular
motion
• An object moving in a circle at a constant
speed still has an acceleration due to its
change in direction
• Velocity is a vector so acceleration can be
produced by a change in magnitude and
direction
• Centripetal Acceleration is acceleration
caused by a change in direction, directed
toward the center of a circular path
• ac = Vt2 / r
Centripetal Acceleration
at and ac
•
•
•
•
Are perpendicular and not the same thing
at is due to changing speed
ac is due to change in direction
To find the total (at & ac) use Pythagorean
theorem
• Direction of total acceleration can be found
using trig functions
Board Work
• A test car moves at a constant speed around a
circular track. If the car is 48.2 m from the
track’s center and has a centripetal acceleration
of 8.05 m/s2, what is the car’s tangential speed?
• The tub of a washing machine has a radius of 34
cm. During the spin cycle, the wall of tub rotates
with a tangential speed of 5.5 m/s. Calculate the
centripetal acceleration of the clothes against
the tub
Causes of Circular Motion
• Centripetal Force: force that maintains
circular motion
• This force is necessary for circular motion
• Ball moving in a circle: Δv due to Δ in
direction
• ac is inward: ac = vt2/r
• Fc is used to change an objects straight
line inertia
• Fc = mac = mvt2 / r
Without a
centripetal force,
an object in
motion
continues along
a straight-line
path.
With a
centripetal
force, an object
in motion will
be accelerated
and change its
direction.
Ff and Fc
• Inertia is often misinterpreted as a force
• Fc is the force directed toward the center and is
necessary for circular motion
• Many times Fc is the force provided by friction
• If the force is lost the object leaves at a tangent
to the circular motion
Force that maintains
circular motion
• A pilot is flying a small plane at 56.6 m/s in
a circular path with a radius of 188.5 m. If
a force of 18,900 N is needed to maintain
the pilot’s circular motion, what is the
plane’s mass?
• A 2000. kg car rounds a circular turn of
radius 20.0 m If the road is flat and the
coefficient of static friction between the
tires and the road is 0.70, how fast can the
car go without skidding?
Gravitational Force
• Orbiting objects are in free fall
• When objects are orbiting, the gravitational
force between the object and Earth is a
centripetal force that keeps the object in orbit
Each successive cannonball
has a greater initial speed, so
the horizontal distance that
the ball travels increases. If
the initial speed is great
enough, the curvature of
Earth will cause the
cannonball to continue falling
without ever landing.
Newton’s Hypothesis
Newton compared motion of the moon to a
cannonball fired from the top of a high mountain.
• If a cannonball were fired with a small
horizontal speed, it would follow a parabolic
path and soon hit Earth below.
• Fired faster, its path would be less curved and
it would hit Earth farther away.
• If the cannonball were fired fast enough, its
path would become a circle and the
cannonball would circle indefinitely.
• This original drawing by
Isaac Newton shows how
a projectile fired fast
enough would fall around
Earth and become an
Earth satellite.
• Both the orbiting cannonball and the moon have
a component of velocity parallel to Earth’s
surface.
• This sideways or tangential velocity is sufficient
to ensure nearly circular motion around Earth
rather than into it.
• With no resistance to reduce its speed, the
moon will continue “falling” around and around
Earth indefinitely.
The gravitational force attracts Earth and the moon to each other.
According to Newton’s 3rd Law.
Newton’s Law of Gravitation
• Gravitational force: the mutual force of
attraction between the particles of matter
• Keeps the planets orbiting around the sun
• Exists between any two masses
regardless of size or composition
• Fg is inversely proportional to distance
• Distance increases, gravity decreases
Newton’s Law of Gravitation
• Fg is localized to
the center of a
spherical mass
• Fg = G (m1m2/r2)
• G is gravitational
constant = 6.673e
-11 Nm2/kg2
Newton’s Law of Gravitation
• Find the Fg exerted on the moon
(m=7.36e22 kg) by Earth (m=5.98e24 kg)
when the distance between them is 3.84e8
m
• Find the distance between a 0.30 kg ball
and a 0.40 kg ball if the magnitude of the
Fg is 8.92e-11 N
Newton’s Law of Universal
Gravitation
• The value of G tells us that gravity is a very
weak force.
• It is the weakest of the presently known four
fundamental forces.
• We sense gravitation only when masses like that
of Earth are involved.
• Cavendish’s first measure of G was called the “Weighing
the Earth” experiment.
• Once the value of G was known, the mass of Earth was
easily calculated.
• The force that Earth exerts on a mass of 1 kilogram at its
surface is 10 newtons.
• The distance between the 1-kilogram mass and the
center of mass of Earth is Earth’s radius, 6.4 × 106
meters.
from which the mass of Earth m1 = 6 × 1024 kilograms.
• When G was first
measured in the 1700s,
newspapers everywhere
announced the discovery
as one that measured the
mass of Earth
Gravitational Field
• We can regard the moon as in contact with the
gravitational field of Earth.
• A gravitational field occupies the space
surrounding a massive body.
• A gravitational field is an example of a force
field, for any mass in the field space experiences
a force.
• Gravitational field strength equals free-fall
acceleration
• Field lines can also represent the pattern of
Earth’s gravitational field.
• The field lines are closer together where the
gravitational field is stronger.
• Any mass in the vicinity of Earth will be
accelerated in the direction of the field lines at
that location.
• Earth’s gravitational field follows the inversesquare law.
• Earth’s gravitational field is strongest near
Earth’s surface and weaker at greater distances
from Earth
• Field lines represent the gravitational field about
Earth
Applications of Gravity
• Weight changes with location
• Gravitational mass equals
inertial mass
• On the surface of any planet,
the value of g, as well as your
weight, will depend on the
planet’s mass and radius
• Your weight is less at the top
of a mountain because you are
farther from the center of
Earth.
• Newton’s law of gravitation accounts for ocean
tides.
• High and low tides are partly due to the
gravitational force exerted on Earth by its moon.
• The tides result from the difference between the
gravitational force at Earth’s surface and at
Earth’s center.
• The moon’s attraction is stronger on Earth’s
oceans closer to the moon, and weaker on the
oceans farther from the moon.
• This is simply because the gravitational force is
weaker with increased distance.
• The two tidal bulges remain relatively fixed with
respect to the moon while Earth spins daily
beneath them.
• Earth’s tilt causes the two daily high tides to be
unequal.
Kepler’s Laws of Planetary
Motion
• Newton’s law of gravitation
was preceded by Kepler’s
laws of planetary motion.
• Kepler’s laws of planetary
motion are three important
discoveries about planetary
motion made by the German
astronomer Johannes
Kepler.
• Kepler started as an assistant to
Danish astronomer Tycho Brahe,
who headed the world’s first great
observatory in Denmark, prior to
the telescope.
• Using instruments called
quadrants, Brahe measured the
positions of planets so accurately
that his measurements are still
valid today.
• After Brahe’s death, Kepler
devoted many years of his life to
the analysis of Brahe’s
measurements.
• Kepler’s laws were developed a
generation before Newton’s law of
universal gravitation.
• Newton demonstrated that Kepler’s laws
are consistent with the law of universal
gravitation.
• The fact that Kepler’s laws closely
matched observations gave additional
support for Newton’s theory of gravitation.
• Kepler’s laws describe the motion of the
planets.
• First Law: Each planet travels in an
elliptical orbit around the sun, and the sun
is at one of the focal points.
• Second Law: An imaginary line drawn
from the sun to any planet sweeps out
equal areas in equal time intervals.
• Third Law: The square of a planet’s orbital
period (T 2) is proportional to the cube of
the average distance (r 3) between the
planet and the sun.
Kepler’s 1st Law
• Kepler’s expectation that
the planets would move in
perfect circles around the
sun was shattered after
years of effort.
• He found the paths to be
ellipses.
Kepler’s 2nd Law
• According to Kepler’s second law, if the time a planet
takes to travel the arc on the left (∆t1) is equal to the time
the planet takes to cover the arc on the right (∆t2), then
the area A1 is equal to the area A2.
• Thus, the planet travels faster when it is closer to the sun
and slower when it is farther away
• After ten years of searching for a connection
between the time it takes a planet to orbit the sun
and its distance from the sun, Kepler discovered
a third law.
• Kepler found that the square of any planet’s
period (T) is directly proportional to the cube of
its average orbital radius (r).
• Kepler’s third law states that T 2  r 3.
• The constant of proportionality is 4p 2/Gm, where
m is the mass of the object being orbited.
Board Work
• Magellan was the first planetary spacecraft
to be launched from a space shuttle.
During the spacecraft’s fifth orbit around
Venus, Magellan traveled at a mean
altitude of 361km. If the orbit had been
circular, what would Magellan’s period and
speed have been?
• Kepler was the first to coin the word satellite.
• He had no clear idea why the planets moved as
he discovered. He lacked a conceptual model.
• Kepler was familiar with Galileo’s concepts of
inertia and accelerated motion, but he failed to
apply them to his own work.
• Like Aristotle, he thought that the force on a
moving body would be in the same direction as
the body’s motion.
• Kepler never appreciated the concept of inertia.
Galileo, on the other hand, never appreciated
Kepler’s work and held to his conviction that the
planets move in circles.
Rotational Motion
• Rotational and translational motion can be
analyzed separately.
• For example, when a bowling ball strikes the
pins, the pins may spin in the air as they fly
backward.
• The pins have both rotational and translational
motion.
• Measure the ability of a force to rotate an object.
Torque
• Torque is a quantity that measures the ability of
a force to rotate an object around some axis.
• How easily an object rotates on both how much
force is applied and on where the force is
applied.
• The perpendicular distance from the axis of
rotation to a line drawn along the direction of the
force is equal to d sin q and is called the lever
arm.
t = Fd sin q
torque = force  lever arm
• The applied force may
act at an angle.
• However, the direction
of the lever arm (d sin
q) is always
perpendicular to the
direction of the applied
force, as shown here.
In each example, the cat is pushing on the door at
the same distance from the axis. To produce the
same torque, the cat must apply greater force for
smaller angles.
• Sign of Torque
• Torque is a vector quantity. In this textbook, we
will assign each torque a positive or negative
sign, depending on the direction the force tends
to rotate an object.
• We will use the convention that the sign of the
torque is positive if the rotation is
counterclockwise and negative if the rotation is
clockwise
Board Work
A basketball is being pushed by two players
during tip-off. One player exerts an upward force
of 15 N at a perpendicular distance of 14 cm
from the axis of rotation. The second player
applies a downward force of 11 N at a distance
of 7.0 cm from the axis of rotation.
Find the net torque acting on
the ball about its
center of mass.
Simple Machines
• A machine is any device that transmits or
modifies force, usually by changing the force
applied to an object.
• All machines are combinations or modifications
of six fundamental types of machines, called
simple machines.
• These six simple machines are the lever, pulley,
inclined plane, wheel and axle, wedge, and
screw
• Because the purpose of a simple machine is to
change the direction or magnitude of an input
force, a useful way of characterizing a simple
machine is to compare the output and input force.
• This ratio is called mechanical advantage.
• If friction is disregarded, mechanical advantage
can also be expressed in terms of input and output
distance.
Fout din
MA 

Fin dout
In the first example, a force
(F1) of 360 N moves the
trunk through a distance
(d1) of 1.0 m. This requires
360 N•m of work.
In the second example, a
lesser force (F2) of only
120 N would be needed
(ignoring friction), but the
trunk must be pushed a
greater distance (d2) of 3.0
m. This also requires 360
N•m of work.
• The simple machines we have considered so far
are ideal, frictionless machines.
• Real machines, however, are not frictionless.
Some of the input energy is dissipated as sound
or heat.
• The efficiency of a machine is the ratio of useful
work output to work input.
Wout
eff 
Win
The efficiency of an ideal
(frictionless) machine is 1, or
100 percent.
The efficiency of real
machines is always less than
1.