### Assignment 8 solution

If you have not done so before, please check the grades at the bottom of the page.
1. Text, problem 9.11. This sort of calculation was done in lecture, but this time think of the vectors in Cartesian coordinates. It is enough to work out just one equation of motion (for x, for example), and recognize this as one component of a vector equation. The same must then be true for the other two, y- and z-components. Can you give a good reason why one equation is enough?
2. Text, problem 9.14
3. Text, problem 9.29. You can also do this "from scratch" arguing entirely in an inertial frame, if you make the appropropriate approximations.
4. Text, problem 1.46. For definiteness let the speed in the puck in S be v, and let S' rotate with angular velocity ω.
(c) Let the orbit in S' be described in Cartesian coordinates (related to polar as in 1.37) by y = a + bx + cx² + dx³ + ... Show that a = b = d = f = ...= 0. Evaluate c and e (and g, ... if you can).
(d) Look at the movie of the demo CORIOLIS EFFECT - BALL ON ROTATING PLATFORM
5.
Foucault pendulum in one simple diagram
We want to derive Eq 9.68, which gives the angular speed at which the plane of oscillation of a pendulum rotates as seen by an observer fixed on the earch (which itself is rotating at the rate Ω). Draw the earth as a circle, N pole on top. Pick a point P between pole and equator, at latitude θ, where the pendulum (oscillator) will be. Draw the "horizontal" (tangent) plane at that point, as a line (unless you are good at 3D drawing), and draw the vector Ω at that chosen point P.
Now consider the vector ω that describes the rotation of the plane of oscillation of the pendulum, as seen from the earth observer. (Since each swing of the pendulum bob is really a slightly curved arc, this "plane of oscillation" is not really plane, but very closely so. In any case it is a vertical plane at P.) Decide what the direction of ω must therfore be; draw that as a line (if you think you should worry about the centrifugal force, don't -- its effect is negligible).
In a frame rotating with that plane, the EOM of the pendulum should be just that of an ordinary pendulum, no Coriolis force. The total angular velocity of that frame, as seen from an inertial frame, is the sum of the earth's rotation with respect to the inertial frame, and the plane rotation with respect to the earth. Keeping that sum in mind look at a Coriolis force equation such as 9.49 and decide what the direction of that sum can be in order that no Coriolis force act on the pendulum. Explain how you decided, draw the vector ω with the correct magnitude, and write an equation for that length. You may also want to comment whether the direction of your ω is reasonable, say by considering what happens if P is at the north pole.

Please check that I have recorded all of your grades correctly. For privacy reasons the list below has no names, and is ordered by the grades on the midterm. This should enable you to find yourself. Let me know of any errors!

Hwk 1    Hwk 2      Hwk 3    Hwk 4        Hwk 5  1st exam    Hwk 6  Hwk 7a

5.5        4                   38
8.7      6          4.5        8          8.5      50                 2
9        8.5        2.5        9          5.5      53        1
9        5.5        7          5.5        5        54
9        7                     7.5        6        57
9        7.5        6.5        6.5        7.5      59        7        1.5
9.3      9          9          9         10        59                 2
9.5      9          7          9          5        60        8
8.5      7.5        5.5        8          8.5      61                 1.5
9.8      7.5                   9          5.5      62        9
8        6.5        3          8.5        6        64        6
9.7      8          6          9.5        5.5      65
9.5      6          5.5        4          5.5      68        8.5
9.3      9          9                              70
9.5                                                71
10       7.5        6.5        9.5        7        73        9.5       2
9.2      5.5        7          7.5        8.5      73                  2
10       7.5       10          9         10        73       10         1.5
8.8      9          7.5        9.5        7.5      78        8         2
8.5      7          8          7.5        7        83        7.5       1.5
9        8          6.5                   6        83        8.5
10       9          6.5        9          6.5      85        9         1
10       8.5        8          8          6.5      88        7.5       1.5
9.5      9          7          9          7.5      89        9         1
8.5      6.5        9.5        8.5        9.5      94       10         2
10      10         10         10         10        96       10         2
10       7.5        9.5        9          9        97        9.5