Lecture 16 Rotational Dynamics. Announcements: Office hours today 1:00 – 3:00.

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Transcript of Lecture 16 Rotational Dynamics. Announcements: Office hours today 1:00 – 3:00.

  • Slide 1
  • Lecture 16 Rotational Dynamics
  • Slide 2
  • Announcements: Office hours today 1:00 3:00
  • Slide 3
  • Angular Momentum
  • Slide 4
  • Consider a particle moving in a circle of radius r, I = mr 2 L = I = mr 2 = rm(r) = rmv t = rp t
  • Slide 5
  • Angular Momentum For more general motion (not necessarily circular), The tangential component of the momentum, times the distance
  • Slide 6
  • Angular Momentum For an object of constant moment of inertia, consider the rate of change of angular momentum analogous to 2nd Law for Linear Motion
  • Slide 7
  • Conservation of Angular Momentum If the net external torque on a system is zero, the angular momentum is conserved. As the moment of inertia decreases, the angular speed increases, so the angular momentum does not change.
  • Slide 8
  • Figure Skater a)the same b)larger because shes rotating faster c) smaller because her rotational inertia is smaller A figure skater spins with her arms extended. When she pulls in her arms, she reduces her rotational inertia and spins faster so that her angular momentum is conserved. Compared to her initial rotational kinetic energy, her rotational kinetic energy after she pulls in her arms must be:
  • Slide 9
  • Figure Skater a)the same b)larger because shes rotating faster c) smaller because her rotational inertia is smaller A figure skater spins with her arms extended. When she pulls in her arms, she reduces her rotational inertia and spins faster so that her angular momentum is conserved. Compared to her initial rotational kinetic energy, her rotational kinetic energy after she pulls in her arms must be: KE rot = I 2 = L (used L = I ). Because L is conserved, larger means larger KE rot. Where does the extra energy come from?
  • Slide 10
  • KE rot = I 2 = L (used L = I ). Because L is conserved, larger means larger KE rot. Where does the extra energy come from? As her hands come in, the velocity of her arms is not only tangential... but also radial. So the arms are accelerated inward, and the force required times the r does the work to raise the kinetic energy
  • Slide 11
  • Conservation of Angular Momentum Angular momentum is also conserved in rotational collisions larger I, same total angular momentum, smaller angular velocity
  • Slide 12
  • Rotational Work A torque acting through an angular displacement does work, just as a force acting through a distance does. The work-energy theorem applies as usual. Consider a tangential force on a mass in circular motion: = r F s = r W = s F Work is force times the distance on the arc: W = (r ) F = rF =
  • Slide 13
  • Rotational Work and Power Power is the rate at which work is done, for rotational motion as well as for translational motion. Again, note the analogy to the linear form (for constant force along motion):
  • Slide 14
  • a) case (a) b) case (b) c) no difference d) it depends on the rotational inertia of the dumbbell A force is applied to a dumbbell for a certain period of time, first as in (a) and then as in (b). In which case does the dumbbell acquire the greater energy ? Dumbbell II
  • Slide 15
  • a) case (a) b) case (b) c) no difference d) it depends on the rotational inertia of the dumbbell A force is applied to a dumbbell for a certain period of time, first as in (a) and then as in (b). In which case does the dumbbell acquire the greater energy ? Dumbbell II If the CM velocities are the same, the translational kinetic energies must be the same. Because dumbbell (b) is also rotating, it has rotational kinetic energy in addition.
  • Slide 16
  • A 2.85-kg bucket is attached to a disk-shaped pulley of radius 0.121 m and mass 0.742 kg. If the bucket is allowed to fall, (a) what is its linear acceleration? (b) What is the angular acceleration of the pulley? (c) How far does the bucket drop in 1.50 s?
  • Slide 17
  • A 2.85-kg bucket is attached to a disk-shaped pulley of radius 0.121 m and mass 0.742 kg. If the bucket is allowed to fall, (a) What is its linear acceleration? (b) What is the angular acceleration of the pulley? (c) How far does the bucket drop in 1.50 s? Pulley spins as bucket falls (c) (b) (a)
  • Slide 18
  • The Vector Nature of Rotational Motion The direction of the angular velocity vector is along the axis of rotation. A right-hand rule gives the sign. Right-hand Rule: your fingers should follow the velocity vector around the circle Optional material Section 11.9
  • Slide 19
  • The Torque Vector Similarly, the right-hand rule gives the direction of the torque vector, which also lies along the assumed axis or rotation Right-hand Rule: point your RtHand fingers along the force, then follow it around. Thumb points in direction of torque. Optional material Section 11.9
  • Slide 20
  • The linear momentum of components related to the vector angular momentum of the system Optional material Section 11.9
  • Slide 21
  • Applied tangential force related to the torque vector Optional material Section 11.9
  • Slide 22
  • Applied torque over time related to change in the vector angular momentum. Optional material Section 11.9
  • Slide 23
  • a) remain stationary b) start to spin in the same direction as before flipping c) start to spin in the same direction as after flipping You are holding a spinning bicycle wheel while standing on a stationary turntable. If you suddenly flip the wheel over so that it is spinning in the opposite direction, the turntable will: Spinning Bicycle Wheel Spinning Bicycle Wheel
  • Slide 24
  • What is the torque (from gravity) around the supporting point? Which direction does it point? Without the spinning wheel: does this make sense? With the spinning wheel: how is L changing? Why does the wheel not fall? Does this violate Newtons 2 nd ?
  • Slide 25
  • Gravity
  • Slide 26
  • Newtons Law of Universal Gravitation Newtons insight: The force accelerating an apple downward is the same force that keeps the Moon in its orbit. Universal Gravitation
  • Slide 27
  • The gravitational force is always attractive, and points along the line connecting the two masses: The two forces shown are an action-reaction pair. If an object is being acted upon by several different gravitational forces, the net force on it is the vector sum of the individual forces. This is called the principle of superposition. G is a very small number; this means that the force of gravity is negligible unless there is a very large mass involved (such as the Earth).
  • Slide 28
  • Gravitational Attraction of Spherical Bodies Gravitational force between a point mass and a sphere * : the force is the same as if all the mass of the sphere were concentrated at its center. a consequence of 1/r 2 (inverse square law) *Sphere must be radial symmetric
  • Slide 29
  • Gravitational Force at the Earths Surface The center of the Earth is one Earth radius away, so this is the distance we use: The acceleration of gravity decreases slowly with altitude......until altitude becomes comparable to the radius of the Earth. Then the decrease in the acceleration of gravity is much larger: g
  • Slide 30
  • In the Space Shuttle Astronauts in the space shuttle float because: a) they are so far from Earth that Earths gravity doesnt act any more b) gravitys force pulling them inward is cancelled by the centripetal force pushing them outward c) while gravity is trying to pull them inward, they are trying to continue on a straight-line path d) their weight is reduced in space so the force of gravity is much weaker
  • Slide 31
  • In the Space Shuttle Astronauts in the space shuttle float because: a) they are so far from Earth that Earths gravity doesnt act any more b) gravitys force pulling them inward is cancelled by the centripetal force pushing them outward c) while gravity is trying to pull them inward, they are trying to continue on a straight-line path d) their weight is reduced in space so the force of gravity is much weaker Astronauts in the space shuttle float because they are in free fall around Earth, just like a satellite or the Moon. Again, it is gravity that provides the centripetal force that keeps them in circular motion. Follow-up: How weak is the value of g at an altitude of 300 km?
  • Slide 32
  • Satellite Motion: F G and a cp Consider a satellite in circular motion * : * not all satellite orbits are circular! Gravitational Attraction: Necessary centripetal acceleration: Does not depend on mass of the satellite! larger radius = smaller velocity smaller radius = larger velocity Relationship between F G and a cp will be important for many gravitational orbit problems
  • Slide 33
  • A geosynchronous satellite is one whose orbital period is equal to one day. If such a satellite is orbiting above the equator, it will be in a fixed position with respect to the ground. These satellites are used for communications and weather forecasting. How high are they? R E = 6378 km M E = 5.87 x 10 24 kg
  • Slide 34
  • Averting Disaster a) its in Earths gravitational field b) the net force on it is zero c) it is beyond the main pull of Earths gravity d) its being pulled by the Sun as well as by Earth e) none of the above The Moon does not crash into Earth because:
  • Slide 35
  • The Moon does not crash into Earth because of its high speed. If it stopped moving, it would, of course, fall directly into Earth. With its high speed, the Moon would fly off into space if it werent for gravity providing the centripetal force. Averting Disaster The Moon does not crash into Earth because: Follow-up: What happens to a satellite orbiting Earth as it slows? a) its in Earths gravitational field b) the net force on it is zero c) it is beyond the main pull of Earths gravity d) its being pulled by the Sun as well as by Earth e) none of the above
  • Slide 36
  • Two Satellites a) 1 / 8 b) c) d) its the same e) 2 Two satellites A and B of the same mass are going around Earth in concentric orbits. The distance of satellite B from Earths center is twice that of satellite A. What is the ratio of the centripetal force acting on B compared to that acting on A?
  • Slide 37
  • Using the Law of Gravitation: we find that the ratio is. Two Satellites a) 1 / 8 b) c) d) its the same e) 2 Two satellites A and B of the same mass are going around Earth in concentric orbits. The distance of satellite B from Earths center is twice that of satellite A. What is the ratio of the centripetal force acting on B compared to that acting on A? Note the 1/R 2 factor
  • Slide 38
  • Gravitational Potential Energy Gravitational potential energy, just like all other forms of energy, is a scalar. It therefore has no components; just a sign. Gravitational potential energy of an object of mass m a distance r from the Earths center: (U =0 at r -> infinity) Very close to the Earths surface, the gravitational potential increases linearly with altitude:
  • Slide 39
  • Energy Conservation Total mechanical energy of an object of mass m a distance r from the center of the Earth: This confirms what we already know as an object approaches the Earth, it moves faster and faster.
  • Slide 40
  • Escape Speed Escape speed: the initial upward speed a projectile must have in order to escape from the Earths gravity from total energy: If initial velocity = v e, then velocity at large distance goes to zero. If initial velocity is larger than v e, then there is non-zero total energy, and the kinetic energy is non-zero when the body has left the potential well
  • Slide 41
  • Maximum height vs. Launch speed Speed of a projectile as it leaves the Earth, for various launch speeds
  • Slide 42
  • Black holes If an object is sufficiently massive and sufficiently small, the escape speed will equal or exceed the speed of light light itself will not be able to escape the surface. This is a black hole. The light itself has mass (in the mass/energy relationship of Einstein), or spacetime itself is curved
  • Slide 43
  • Gravity and light Light will be bent by any gravitational field; this can be seen when we view a distant galaxy beyond a closer galaxy cluster. This is called gravitational lensing, and many examples have been found.
  • Slide 44
  • Keplers Laws of Orbital Motion Johannes Kepler made detailed studies of the apparent motions of the planets over many years, and was able to formulate three empirical laws You already know about circular motion... circular motion is just a special case of elliptical motion 1. Planets follow elliptical orbits, with the Sun at one focus of the ellipse. Only force is central gravitational attraction - but for elliptical orbits this has both radial and tangential components Elliptical orbits are stable under inverse-square force law.
  • Slide 45
  • Keplers Laws of Orbital Motion 2. As a planet moves in its orbit, it sweeps out an equal amount of area in an equal amount of time. This is equivalent to conservation of angular momentum v t r
  • Slide 46
  • Keplers Laws of Orbital Motion 3. The period, T, of a planet increases as its mean distance from the Sun, r, raised to the 3/2 power. This can be shown to be a consequence of the inverse square form of the gravitational force.
  • Slide 47
  • Orbital Maneuvers Which stable circular orbit has the higher speed? How does one move from the larger orbit to the smaller orbit?
  • Slide 48
  • Binary systems If neither body is infinite mass, one should consider the center of mass of the orbital motion
  • Slide 49
  • If you weigh yourself at the equator of Earth, would you get a bigger, smaller, or similar value than if you weigh yourself at one of the poles? a) bigger value b) smaller value c) same value Guess My Weight
  • Slide 50
  • If you weigh yourself at the equator of Earth, would you get a bigger, smaller, or similar value than if you weigh yourself at one of the poles? a) bigger value b) smaller value c) same value normal force you are in circular motion net inward forcenormal force must be slightly less than mg The weight that a scale reads is the normal force exerted by the floor (or the scale). At the equator, you are in circular motion, so there must be a net inward force toward Earths center. This means that the normal force must be slightly less than mg. So the scale would register something less than your actual weight. Guess My Weight
  • Slide 51
  • Earth and Moon I a) the Earth pulls harder on the Moon b) the Moon pulls harder on the Earth c) they pull on each other equally d) there is no force between the Earth and the Moon e) it depends upon where the Moon is in its orbit at that time Which is stronger, Earths pull on the Moon, or the Moons pull on Earth?
  • Slide 52
  • By Newtons Third Law, the forces are equal and opposite. Earth and Moon I a) the Earth pulls harder on the Moon b) the Moon pulls harder on the Earth c) they pull on each other equally d) there is no force between the Earth and the Moon e) it depends upon where the Moon is in its orbit at that time Which is stronger, Earths pull on the Moon, or the Moons pull on Earth?