Conservation of Angular Momentum and Kepler's Laws in Physics

1. Schedule • Today, April 25 • Conservation of Angular Momentum and Special Topics (11-7, 11-8, 11-9, 11-10) • Friday, April 27 • Static Equilibrium (12-1, 12-2, 12-3, 12-4) • Quiz 4 Review • Monday, April 30 • Quiz 4 / 50 points / Crib Sheet • Wednesday, May 2 • Catch-up / Final Review • Wednesday, May 9 noon-1:50PM • Final / 100 points / Open book Physics 253

2. From Gravity Probe B: mass and rotation of earth change direction of space/time http://einstein.stanford.edu/ The Gyroscope • Remember our asymmetric mass? We found that a symmetric body requires no torque to keep it spinning about the axis of symmetry. • A rapidly spinning wheel or sphere mounted so that it is essentially frictionless provides an infallible sense of direction. • Since it is symmetric the bearings introduce no torques And the axis of rotation does not change no matter what happens to the mount. • This is used for navigation in aircraft and satellites. But it can be used in experiments by observing a distant star, and shifts are due to the reference frame moving! Perfect niobium spheres Physics 253

3. Conservation of Angular Momentum • In Chapter 9 we saw that Newton’s 2nd Law SF=ma could be rewritten as SF=dP/dt for both a single particle or a system of particles. • When the net forces are zero, dP/dt=0 or P=constant, this is the Conservation Law of Momentum. • Similarly St=dL/dt is true for a system of particles in an inertial frame or about the center of mass. • If the net torques are zero then dL/dt=0 and L=constant, which is the powerful Conservation Law of Angular Momentum. Physics 253

4. Deriving Kepler’s 2nd Law • Three Laws • 1st Law: Path is an ellipse (for physics majors!) • 2nd Law: Equal Areas in equal time. • 3rd Law: Ratio of squares of the periods of any two planets equal to the ratio of the cubes of semi-major axes (already proven with the universal law of gravitation!) • We can use the conservation of angular momentum to show that each planet moves such that a line from the Sun to the planet sweeps out equal areas in equal periods of time. That is, prove 2nd Law Physics 253

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6. Another Example: A Ballistic Cylinder • A bullet of mass m and velocity v strikes and becomes embedded in the edge of a cylinder of mass M and radius R. • The cylinder initially at rest begins to rotate about it’s stationary symmetry axis. What is the angular velocity of the cylinder after the collision? Physics 253

7. Note that the angular velocity increases linear with velocity and bullet mass and is inversely related to cylinder mass and radius. Physics 253

8. Imagine a symmetric top of mass M rapidly spinning around its symmetry axis with its tip at the origin O in an inertial reference frame. The symmetry axis is tipped at an angle f with respect to vertical. When the top is released rather than just fall down (as it would if not spinning) the axis will move around the vertical axis. Such movement is called precession, with an angular velocity of precession W. We now know enough now to analyze and explain this amazing and complicated motion! The Spinning Top! Physics 253

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11. Amazingly the rate of precession does not depend on the tilt angle. But it does depend on: the height of the CM (r), the mass and shape of the top (M,I), the acceleration of gravity (g), and the rotation speed (w) (as w or L increases precession slows down!) Physics 253

12. Rotating Frames of Reference • Frames of reference matter! Our usual frame of reference is the surface of the earth. But actually this is a rotating frame and not truly inertial. • We actually get a different view if we put ourselves above the earth, then we see that a frame on the earth’s is clearly rotating, so is accelerating, and is not inertial. • But then… we still aren’t in an inertial frame since we are rotating about the sun, and the sun about the galaxy, etc… • For most purposes (like a falling ball) all of these accelerations matter little since their effects are small over the time scales involved. • But there are everyday consequences, let’s see how… Physics 253

13. Altered Views • We can learn a lot by placing ourselves into a reference frame that’s rotating. • Imagine sitting on a rotating platform. The two possible view points have extremely different aspects: • First of all, it looks like the world is going around YOU (just like the stars do)! • But of course you are spinning around a fixed point. Physics 253

14. Let’s consider the motion of a ball set down on the surface of the frictionless platform. • The person on the platform will see the ball accelerating and moving off the platform. This clearly is in conflict with Newton’s 2nd Law, the ball moves from rest in the absence of a force. A big mystery to a Newtonian rider and an example of a non-inertial force and frame! • A 2nd person looking down from above would have a clear view: the ball has an initial tangential velocity v=rw when set down and continues to move in a straight line. This is in accordance with Newton’s 2nd law and so the 2nd person is in an inertial frame. Physics 253

15. Handling Rotating Frames • Since a rotating frame is non-inertial it seems we can’t use Newton’s Laws in this frame. • But we can finesse the problem by introducing fictional or non-inertial forces. Consider the ball: • From the viewer above, if it was to stay on the merry-go-round at the same position, there must be a force mv/r2 holding it in place. • But on the merry-go-round there seems to be a force driving it away. The magnitude of that force must also be mv/r2 and outward – the so called centrifugal force. This is an example of a fictitious force also called an inertial force or a pseudo-force. • If we assume it acts on all objects in the non-inertial frame then Newton’s Laws can be applied. Although conceptually straightforward, the form of this pseudo-force inertial force is a bit complicated. Let’s investigate… Physics 253

16. The Coriolis Effect • In a reference frame rotating at a constant w a pseudo-force appears to asserts itself on object in motion in the rotating reference frame. • Consider a thrown ball, an airplane, or parcels of air in a hurricane. • This pseudo-force or inertial force acts to deflect the object sideways or at right angles to the motion and is known as the Coriolis force. Gaspard Gustave de Coriolis 1792 – 1843 a French Engineer/Mathematician Physics 253

17. Consider two people at positions A and B on a rotating platform and at rA and rB. The innermost person at A throws a ball outward to the person at B. • In the inertial frame or the “frame from above” • the ball has two velocity components, the first v due to the throw and the second the tangential velocity or vA=rAw. • However the outermost person has a greater tangential velocity vB=rBw and he will “race ahead” of the ball and it will pass behind him/her. • In the rotating reference frame the ball will be still not reach the person and seems to be accelerated sideways by a force called the Coriolis force. This is due entirely to the rotating frame and the fact that as the radii increase the tangential velocity increases. Physics 253

18. The Rotation of Storms! • Explained by the Coriolis force. • Just project the northern hemisphere onto a rotating disk, the higher latitudes have smaller radii and smaller tangential velocity, the lower latitudes have greater tangential velocity. • Just as with the ball on the rotating platform as a parcel of air rushes from the north towards the low pressure we perceive it to be deflected to the right or west. • As a parcel of air rushes from the south towards the low pressure it’s also deflected to the right or to the east. • Viola, counter-clockwise rotation! Just due to the different tangential velocities at each latitude. towards the east Physics 253

19. With this simple picture we can actually calculate the magnitude of the acceleration as viewed from the rotating frame. • In the inertial frame or “frame from above” the radial motion of the ball is given by: rB-rA=vt. • During this time it moves to the side by sA=vAt. Likewise the person at point B moves to the side sB=vBt. • As a result the ball passes behind the outermost person by the distance: s=sB-sA=(vB-vA)t=(rB-rA)wt. • We can combine the equations in red and find s=wvt2. This is the same sideways shortfall seen in the inertial or rotating frame. • But this can be compared to the equation of motion for constant acceleration y=(1/2)at2, indicating the Coriolis acceleration is 2wv. Physics 253

20. Properties of the Coriolis Acceleration • The acceleration is acor=2wv and is valid for any velocity or velocity component in the plane of rotation. • Note the dependence on v, as winds intensify near the eye of a hurricane the deflection increases, tightening the spiral. • The full result is actually given by the cross product acor=-2wxv where w and v are the angular frequency and v the velocity in the rotating reference frame. • The magnitude is then really 2wvperp where vperp is the component of velocity perpendicular to the rotation axis. Physics 253

21. A ball is dropped vertically from a height h=110m at a latitude of 44 degrees in the Northern Hemisphere. What is the apparent deflection due to the Coriolis force? Well first off since the ball is falling from a larger to a smaller radius it will have greater tangential velocity than the surface. That means it will tend to rotate faster and be deflected in the direction of rotation or to the east. From the figure we can also see that the component of the balls velocity perpendicular to w as it falls is vcosl where l is the latitude. The Deflection of a Falling Ball! Physics 253

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23. Note 1) the faster the rotation the greater the deflection 2) at the equator cos(0)=1 and the effect is maximal, 3) at the pole cos(90)=0 and there is no effect. 4) the greater the height the greater the deflection, for 10000m, x~15m and not to be neglected! Physics 253