Rotational Motion and The Law of Gravity

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Rotational Motion and The Law of Gravity

• Ch 7

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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 axisthat
  is, an axis located within the body of the
  objectthe motion is called rotation, or spin.
• When an object turns about an external axis, the
  motion is called revolution.

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• 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.

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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

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Centripetal Acceleration

• Tangential speed (vt) depends on distance
• When tangential speed is constant, motion is
  described as uniform circular motion

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• 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

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Centripetal Acceleration
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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

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Board Work

• A test car moves at a constant speed around a
  circular track. If the car is 48.2 m from the
  tracks center and has a centripetal acceleration
  of 8.05 m/s2, what is the cars 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

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Causes of Circular Motion

• Centripetal Force force that maintains circular
  motion
• This force is necessary for circular motion
• Ball moving in a circle ?v due to ? in direction
• ac is inward ac vt2/r
• Fc is used to change an objects straight line
  inertia
• Fc mac mvt2 / r

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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.
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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

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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
  pilots circular motion, what is the planes
  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?

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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.
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Newtons 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.

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• This original drawing by Isaac Newton shows how a
  projectile fired fast enough would fall around
  Earth and become an Earth satellite.

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• Both the orbiting cannonball and the moon have a
  component of velocity parallel to Earths
  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.

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The gravitational force attracts Earth and the
moon to each other. According to Newtons 3rd
Law.
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Newtons 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

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Newtons 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

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Newtons Law of Gravitation

• Find the Fg exerted on the moon (m7.36e22 kg) by
  Earth (m5.98e24 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

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Newtons 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.

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• Cavendishs 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 Earths radius, 6.4
  106 meters.
• from which the mass of Earth m1 6 1024
  kilograms.

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• When G was first measured in the 1700s,
  newspapers everywhere announced the discovery as
  one that measured the mass of Earth

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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

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• Field lines can also represent the pattern of
  Earths 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.
• Earths gravitational field follows the
  inverse-square law.
• Earths gravitational field is strongest near
  Earths surface and weaker at greater distances
  from Earth

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• Field lines represent the gravitational field
  about Earth

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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 planets
  mass and radius
• Your weight is less at the top of a mountain
  because you are farther from the center of Earth.

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• Newtons 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 Earths surface and at
  Earths center.
• The moons attraction is stronger on Earths
  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.

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• The two tidal bulges remain relatively fixed with
  respect to the moon while Earth spins daily
  beneath them.

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• Earths tilt causes the two daily high tides to
  be unequal.

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Keplers Laws of Planetary Motion

• Newtons law of gravitation was preceded by
  Keplers laws of planetary motion.
• Keplers laws of planetary motion are three
  important discoveries about planetary motion made
  by the German astronomer Johannes Kepler.

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• Kepler started as an assistant to Danish
  astronomer Tycho Brahe, who headed the worlds
  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 Brahes death, Kepler devoted many years of
  his life to the analysis of Brahes measurements.

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• Keplers laws were developed a generation before
  Newtons law of universal gravitation.
• Newton demonstrated that Keplers laws are
  consistent with the law of universal gravitation.
• The fact that Keplers laws closely matched
  observations gave additional support for Newtons
  theory of gravitation.

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• Keplers 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 planets orbital
  period (T 2) is proportional to the cube of the
  average distance (r 3) between the planet and the
  sun.

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Keplers 1st Law

• Keplers 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.

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Keplers 2nd Law

• According to Keplers 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

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• 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 planets
  period (T) is directly proportional to the cube
  of its average orbital radius (r).
• Keplers 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.

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Board Work

• Magellan was the first planetary spacecraft to be
  launched from a space shuttle. During the
  spacecrafts fifth orbit around Venus, Magellan
  traveled at a mean altitude of 361km. If the
  orbit had been circular, what would Magellans
  period and speed have been?

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• 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 Galileos 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
  bodys motion.
• Kepler never appreciated the concept of inertia.
  Galileo, on the other hand, never appreciated
  Keplers work and held to his conviction that the
  planets move in circles.

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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.

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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

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• 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.

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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.
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• 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

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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.

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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

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• 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.

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In the first example, a force (F1) of 360 N moves
the trunk through a distance (d1) of 1.0 m. This
requires 360 Nm 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 Nm of work.
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• 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.

The efficiency of an ideal (frictionless) machine
is 1, or 100 percent. The efficiency of real
machines is always less than 1.