Circular Motion

1
Circular Motion

• Chapters 9, 12, 13 14 Notes

2
Rotation and Revolution

• Rotation (spin) - occurs when an object spins
  about an internal axis
• Revolution occurs when an object turns about an
  external axis
• The Earth rotates and revolves
• Rotates around its axis (once every 24 hours)
• http//bestanimations.com/EarthSpace/Earth/Earth-
  12-june.gif
• Revolves around the sun (once every 365.25 days)
• http//www.planets.ndo.co.uk/images/ani_earth.gif

3
Rotational Tangential Speed

• Rotational Speed the number of rotations an
  object makes per unit of time
• Linear Speed distance moved per unit of time
• Tangential Speed the linear speed of something
  moving in a circular path
• The direction of motion is always tangent to the
  circle

4
Rotational Tangential Speed

• Rotational Speed of rotations or revolutions
  / time interval
• Find the rotational speed of a carousel that can
  make 2 complete revolutions in 30 s.
• Tangential Speed distance traveled / time
• For one rotation, the distance an object moving
  in a circle travels is the circumference of the
  circle.
• The period of an object is the time it takes to
  make one rotation or revolution.
• The frequency of an object is the number of
  rotations or revolutions an object makes in a
  given unit of time.
• Tangential speed 2pr/T
• Find the tangential speed of a horse that is 1.0
  m from the axis of rotation of a carousel that
  can make 2 complete revolutions in 30 s.

5
Finding the Speed of the Earth

• The radius of the Earth is 6.4 x 106 m.
• Find the rotational speed of a person standing on
  the surface of the Earth.
• Find the tangential speed of a person standing at
  the North Pole.
• Find the tangential speed of a person standing at
  the equator.
• Rotational speed is not dependent upon
  radiusBloomington rotates as fast as Hawaii.
• Tangential speed depends on radiussomeone in
  Bloomington is closer to the axis of rotation
  than someone in Hawaii in Bloomington there is a
  smaller tangential speed.

6
Finding the Speed of the Earth

• The average distance from the Earth to the Sun is
  1.5 x 1011 m.
• Find the tangential speed of the Earth as it
  orbits the Sun.
• If the force of gravity between the Earth and the
  Sun suddenly disappeared, what path would the
  Earth travel?

7
Forced to Go in Circles

• Centripetal Force any force that caused a body
  to move in a circular path or part of a circular
  path

8
Centripetal Acceleration

• If an object is moving in a circle at a constant
  speed, how can there be a force acting on it if
  Newtons 2nd Law (F ma) is true?
• The magnitude of the velocity is constant, but
  the object is constantly changing direction.
• Since velocity is a vector, if the direction
  changes, the velocity changes.
• The object is experiencing centripetal
  acceleration.
• acentripetal v2/r

9
Centripetal Force

• For questions 1-4 An object is moving in a
  clockwise direction around a circle at constant
  speed. Use your understanding of the concepts of
  velocity and acceleration to answer the next four
  questions. Use the diagram shown at the right.
• Which vector below represents the direction of
  the velocity vector when the object is located at
  point B on the circle?
• Which vector below represents the direction of
  the acceleration vector when the object is
  located at point C on the circle?
• Which vector below represents the direction of
  the velocity vector when the object is located at
  point C on the circle?
• Which vector below represents the direction of
  the acceleration vector when the object is
  located at point A on the circle? 

10
Centripetal Force

• A car travels through a valley at constant speed,
  though not at constant velocity.
•  
• How is this possible? Is the car accelerating?
  What direction is the car's acceleration?
• Are the forces balanced? Justify the relative
  sizes of the forces.
• If the car's speed is 25 m/s, its mass is 1200 kg
  and the radius of valley (r) is 25 meters,
  determine the magnitude of the centripetal force
  acting on the car.        

11
Centripetal Force

• Consider your motion on a ferris wheel. The
  radius is 12 meters and the rotational period is
  50 seconds. Use 70 kg for the mass. A force
  diagram should be part of each solution.
• a) How heavy do you feel at the top?
• b) How heavy do you feel at the bottom?
• c) How heavy do you feel on the side?

12
Centripetal Force vs. Centrifugal Force

• Centripetal and centrifugal force are different
  concepts.
• Centripetal force is the name for the "net force
  toward the center of the circle" that causes
  circular motion to occur.
• Centrifugal force is the name for the apparent
  force that "pushes" objects away from the center
  of a circle from the viewpoint of the object that
  is undergoing circular motion. Of course, there
  is no force pushing an object away from the
  center of the circle, it is simply the inertia of
  the object carrying it in a straight line,
  tangent to the circle.
• What type of force do clothes in the spin cycle
  of a washing machine experience?

13
The Falling Apple

• Circular motion (as well as elliptical motion)
  requires a centripetal force.
• The nature of such a force - its cause and its
  origin - bothered Newton
• According to legend, a breakthrough came at age
  24 in an apple orchard in England. Newton never
  wrote of such an event, yet it is often claimed
  that the notion of gravity as the cause of all
  heavenly motion was instigated when he was struck
  in the head by an apple while lying under a tree
  in an orchard in England.
• Whether it is a myth or a reality, the fact is
  certain that it was Newton's ability to relate
  the cause for heavenly motion (the orbit of the
  moon about the earth) to the cause for Earthly
  motion (the fall of an apple to the Earth) which
  led him to his notion of universal gravitation.

14
The Falling Moon

• Newton compared the falling apple to the moon
• If the moon did not fall it would move off in a
  straight line
• The moon is falling around the Earth
• Falls beneath a tangent line drawn to the Earth
  at any given point
• The moon is attracted to the Earth and the Earth
  is attracted to the moon
• All objects in the universe attract one another
• Gravitation is UNIVERSAL

15
The Falling Moon

• Newton explained the concept of the falling moon
  by comparing it to a cannonball fired from the
  top of a mountain
• If the cannonball were fired with a small
  horizontal speed, it would follow a parabolic
  path and would soon hit the
  Earth (path A) http//www.physicsclassroom.com/mme
  dia/vectors/4kms.gif
• If it were fired faster, its path would be less
  curved and it would hit the Earth farther away
  (path B) http//www.physicsclassroom.com/mmedia/ve
  ctors/6kms.gif
• If the cannonball were fired fast enough, its
  path would become a circle and the cannonball
  would circle the Earth (path C)
  http//www.physicsclassroom.com/mmedia/vectors/co.
  gif
• If the cannonball were fired even faster it would
  follow an elliptical path around the Earth (path
  E) http//www.physicsclassroom.com/mmedia/vectors/
  eo.gif

16
Universal Gravitation

• Law of Universal Gravitation
• F Gm1m2/d2
• G 6.67 x 10-11 Nm2/kg2
• G the universal gravitational constant
• What happens to the force if
• the masses increase?
• the distance between the masses increases?

17
Law of Universal Gravitation

• If all bodys are attracted, why arent you
  pulled toward a large building?
• Is your weight at the top of the Sears Tower in
  Chicago more, less, or equal to your weight on
  the street below?

18
Universal Gravitation

• The earth's orbit around the sun is very nearly
  circular, with an average radius of 1.5 x 108 km.
  Assume the mass of the earth is 6 x 1024 kg and
  the mass of the sun is 1.99 x 1030 kg.
• Determine the gravitational force on the earth by
  the sun. How does the force on the earth by the
  sun compare to the force on the sun by the earth?
• What is the magnitude of the earth's average
  acceleration in its orbit around the sun?
• Determine the average speed of the earth in its
  orbit around the sun.

19
Gravitational Fields

• Force Field exerts a force on objects in its
  vicinity
• magnetic fields, electric fields, gravitational
  fields
• Gravitational Field the type of force field
  that surrounds massive objects
• g (the gravitational field vector) depends on two
  things
• Mass
• radius
• The gravitational field is a vector with a
  magnitude of g that points in the direction of
  the gravitational force
• g Fg/m

20
Earths Gravitational Field Strength

• Find the gravitational field strength the Earth
  exerts on a 50 kg person standing on its surface.
  The mass of the Earth is 6 x 1024 and the radius
  is 6.4 x 106 m.

21
Gravitational Fields

• If you dropped a rock into a tunnel through the
  Earth, what would happen?
• It would gain speed until it reached the Earths
  center, and then lose speed the rest of the way.
  Its speed at the far end of the tunnel would be
  the same as its initial speed. It would then
  fall back and repeat the motion in cyclic
  fashion.
• The rock would make a round trip in approximately
  90 minutes (the same time that it takes a close
  orbit satellite to orbit the Earth)
• As the velocity of the rock increases when
  falling into the Earth tunnel, what happens to
  the acceleration?
• It decreases as the gravitational field
  decreases, and is zero at the Earths center.
  The falling body has its maximum velocity at the
  Earths center, where both the field and the
  acceleration are zero.

22
Satellites

• An Earth satellite is a satellite that falls
  around the Earth not into it.
• Gravity keeps a satellite in orbit
• Without gravity, the satellite would keep moving
  in a straight line
• How fast must a stone be thrown to horizontally
  orbit the Earth?
• 8 km/s (29,000 km/h 18,000 mi/h)
• The geometric curvature of the Earth is 5 m
  vertical for every 8 km horizontal
• An object in free fall falls 5 m in 1 s
• In order to match the curve of the Earth, the
  object must travel 8 km horizontally in 1 s

23
Earth Satellites

• The orbital speed for a close orbit Earth
  satellite is 8 km/s
• A close orbit satellite must be at least 150 km
  above the Earths surface
• If it wasnt it would burn up against the
  friction of the atmosphere (like a falling star)

24
Circular Orbits
Is the force constant? Is the acceleration
constant? Is the velocity constant? Is the speed
constant?
25
Circular Orbits

• It takes about 90 minutes for a satellite close
  to the Earth to complete it orbit
• For higher altitudes the speed is less and the
  time of orbit (period) is longer
• The moon is a satellite of Earth and its period
  is 27.3 days
• Communications satellites have a period of 24
  hours (so they are always above the same point on
  the Earth).
• How far away from the center of the Earth is a
  satellite that has a period of 24 hours?

26
Elliptical Orbits

• Is the force constant?
• Is the acceleration constant?
• Is the velocity constant?
• Is the speed constant?

27
Satellite Motion Conservation of Energy

• Moving objects have KINETIC energy
• Objects above the Earths surface have POTENTIAL
  energy
• A satellite has both KE and PE
• The sum of the KE and PE everywhere is constant
• Energy is conserved

28
Conservation of Energy Satellite Motion

• Circular Orbitthe distance from the Earth does
  not change
• PE is constant
• Energy is conservedKE is constant
• If KE is constantvelocity is constant
• Elliptical Orbitthe distance from the Earth and
  the speed of the satellite changes
• PE is greatest where the satellite is farthest
  away (at the apogee)
• KE is smallest where the satellite is farthest
  away (apogee)the satellite is moving at its
  slowest speed
• PE is smallest where the satellite is closest
  (perigee)
• KE is greatest where the satellite is closest
  (perigee)the satellite is moving at its fastest
  speed

29
Energy Conservation Satellite Motion

• Where does the satellite have
• a. Maximum speed
• b. Maximum velocity
• c. Maximum gravitational attraction to the Earth
• d. Maximum kinetic energy
• e. Maximum gravitational potential energy
• f. Maximum total energy
• g. Maximum acceleration
• h. Minimum speed
• i. Increasing speed
• j. Decreasing speed
• k. Force perpendicular to its velocity