Rotational Dynamics, cont. Lecture 16 Rotational Dynamics, cont.

Reading and Review

Dumbbell I 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 center-of-mass speed ? a) case (a) b) case (b) c) no difference d) it depends on the rotational inertia of the dumbbell Answer: c

Dumbbell I 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 center-of-mass speed ? a) case (a) b) case (b) c) no difference d) it depends on the rotational inertia of the dumbbell Because the same force acts for the same time interval in both cases, the change in momentum must be the same, thus the CM velocity must be the same.

Tipping Over I a) all b) 1 only c) 2 only d) 3 only e) 2 and 3 A box is placed on a ramp in the configurations shown below. Friction prevents it from sliding. The center of mass of the box is indicated by a blue dot in each case. In which case(s) does the box tip over? Answer: d 1 2 3

Tipping Over I a) all b) 1 only c) 2 only d) 3 only e) 2 and 3 A box is placed on a ramp in the configurations shown below. Friction prevents it from sliding. The center of mass of the box is indicated by a blue dot in each case. In which case(s) does the box tip over? The torque due to gravity acts like all the mass of an object is concentrated at the CM. Consider the bottom right corner of the box to be a pivot point. Which direction does the torque rotate the box? 1 2 3 If the box can rotate such that the CM is lowered, it will!!

F = ma implies Newton’s first law: without a force, there is no acceleration Now we have Linear momentum was the concept that tied together Newton’s Laws, is there something similar for rotational motion?

Angular Momentum Consider a particle moving in a circle of radius r, I = mr2 L = Iω = mr2ω = rm(rω) = rmvt = rpt

Angular Momentum For more general motion (not necessarily circular), The tangential component of the momentum, times the distance

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

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.

Conservation of Angular Momentum Angular momentum is also conserved in rotational collisions

Figure Skater a) the same b) larger because she’s rotating faster 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: a) the same b) larger because she’s rotating faster c) smaller because her rotational inertia is smaller

Figure Skater a) the same b) larger because she’s rotating faster 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: a) the same b) larger because she’s rotating faster c) smaller because her rotational inertia is smaller KErot = I 2 = L2 / I (used L = I ). Because L is conserved, smaller I means larger KErot. The “extra” energy comes from the work she does on her arms.

Rotational Work s = r Δθ W = (r Δθ) F = rF Δθ = τ Δθ τ = r F A torque acting through an angular displacement does work, just as a force acting through a distance does. Consider a tangential force on a mass in circular motion: τ = r F Work is force times the distance on the arc: s = r Δθ W = s F W = (r Δθ) F = rF Δθ = τ Δθ The work-energy theorem applies as usual.

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):

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

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: your fingers should follow the force vector around the circle

The linear momentum of components related to the vector angular momentum of the system

Applied tangential force related to the torque vector

Applied torque over time related to change in the vector angular momentum.

Cassette Player When a tape is played on a cassette deck, there is a tension in the tape that applies a torque to the supply reel. Assuming the tension remains constant during playback, how does this applied torque vary as the supply reel becomes empty? a) torque increases b) torque decreases c) torque remains constant Answer: b

Cassette Player When a tape is played on a cassette deck, there is a tension in the tape that applies a torque to the supply reel. Assuming the tension remains constant during playback, how does this applied torque vary as the supply reel becomes empty? a) torque increases b) torque decreases c) torque remains constant a) torque increases b) torque decreases c) torque remains constant Answer: b As the supply reel empties, the lever arm decreases because the radius of the reel (with tape on it) is decreasing. Thus, as the playback continues, the applied torque diminishes.

http://arxiv.org/abs/1010.3279

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

The linear momentum of components related to the vector angular momentum of the system Optional material Section 11.9

Spinning Bicycle Wheel 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: 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

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 Newton’s 2nd?

Gravity

Newton’s Law of Universal Gravitation Newton’s insight: The force accelerating an apple downward is the same force that keeps the Moon in its orbit. Universal Gravitation

The two forces shown are an action-reaction pair. The gravitational force is always attractive, and points along the line connecting the two masses: The two forces shown are an action-reaction pair. 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). 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.

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/r2 (inverse square law) *Sphere must be radial symmetric

Gravitational Force at the Earth’s Surface The center of the Earth is one Earth radius away, so this is the distance we use: g 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:

In the Space Shuttle a) they are so far from Earth that Earth’s gravity doesn’t act any more b) gravity’s 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: Answer: c

In the Space Shuttle a) they are so far from Earth that Earth’s gravity doesn’t act any more b) gravity’s 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: 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?

Satellite Motion: FG and acp Consider a satellite in circular motion*: Gravitational Attraction: Necessary centripetal acceleration: Does not depend on mass of the satellite! larger radius = smaller velocity smaller radius = larger velocity Relationship between FG and acp will be important for many gravitational orbit problems * not all satellite orbits are circular!

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 and weather forecasting.

Averting Disaster a) it’s in Earth’s gravitational field b) the net force on it is zero c) it is beyond the main pull of Earth’s gravity d) it’s being pulled by the Sun as well as by Earth e) none of the above The Moon does not crash into Earth because: Answer: e

Averting Disaster a) it’s in Earth’s gravitational field b) the net force on it is zero c) it is beyond the main pull of Earth’s gravity d) it’s being pulled by the Sun as well as by Earth e) none of the above The Moon does not crash into Earth because: 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 weren’t for gravity providing the centripetal force. Follow-up: What happens to a satellite orbiting Earth as it slows?

Two Satellites a) 1/8 b) ¼ c) ½ d) it’s 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 Earth’s center is twice that of satellite A. What is the ratio of the centripetal force acting on B compared to that acting on A? Answer: b

Two Satellites Note the 1/R2 factor a) 1/8 b) ¼ c) ½ d) it’s 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 Earth’s center is twice that of satellite A. What is the ratio of the centripetal force acting on B compared to that acting on A? Using the Law of Gravitation: we find that the ratio is . Note the 1/R2 factor

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 Earth’s center: (U =0 at r -> infinity) Very close to the Earth’s surface, the gravitational potential increases linearly with altitude:

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.

Escape Speed Escape speed: the initial speed a projectile must have in order to escape from the Earth’s gravity from total energy: Must initial velocity be upwards? If we neglect air resistance, then no, it just has to avoid hitting the ground! If initial velocity = ve, then velocity at large distance goes to zero. If initial velocity is larger than ve, then there is non-zero total energy, and the kinetic energy is non-zero when the body has left the potential well

Maximum height vs. Launch speed Speed of a projectile as it leaves the Earth, for various launch speeds

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. Object bright due to radiation from near “event horizon.” The light itself has mass (in the mass/energy relationship of Einstein), or spacetime itself is curved

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.

General Relativity Previous effects examples Clocks run slower in gravitational field - must be taken into account for GPS systems to work - was missed in determining time for neutrinos to go from CERN to Gran Sasso!

Kepler’s 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 1. Planets follow elliptical orbits, with the Sun at one focus of the ellipse. Elliptical orbits are stable under inverse-square force law. You already know about circular motion... circular motion is just a special case of elliptical motion Only force is central gravitational attraction - but for elliptical orbits this has both radial and tangential components

Kepler’s 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. r v Δt This is equivalent to conservation of angular momentum

Kepler’s 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.

Guess My Weight 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 Answer: b

Guess My Weight 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 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 Earth’s center. This means that the normal force must be slightly less than mg. So the scale would register something less than your actual weight.

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, Earth’s pull on the Moon, or the Moon’s pull on Earth? Answer: c

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, Earth’s pull on the Moon, or the Moon’s pull on Earth? By Newton’s Third Law, the forces are equal and opposite.