Physics 272: Advanced Honors Physics II
Spring 2018

HW1 due in class Monday, January 22, 2018
Reading: Purcell 1.1-1.5


Note: the solution for each problem should be written on a separate page (sorry, trees!) to facilitate the grading.
IN ADDITION, YOU MUST COMPLY WITH THE PROBLEM SOLVING CHECKLIST (copy provided for your convenience below)
I have put the answers at the bottom of this page. You can use them to check your own answers -- if they don't agree, you know you need to go back and look for at least one mistake.

1. *Purcell 1.36: repelling volley balls (Coulomb's law, superposition, vector addition of forces, symmetry, trigonometry)
for your convenience in finding the final number: g = 9.8 m/s2 and k = 9.0 x 109 Nm2/C2

2. **Purcell 1.38: small oscillations on a line.
We learned how to do small oscillation problems in 271 last fall. If you are new to the course and don't know how to do these, or need a review, see the help copied below the Problem Solving Checklist and/or come to the Sunday evening office hour.
For your convenience, I am reminding you of the relevant Taylor expansions around x=0
(
1+x)-1 = 1-x+x2+... (this one is used in the energy approach)
(1+x)-2 = 1-2x+...     (this one is used in the force approach)
valid when x is much smaller than 1

3. **Purcell 1.39: potential energy
NOTE: the "energy method" is easier than the "force method" for this problem (write the potential energy U as a function of theta, and then find the value of theta that makes dU/dtheta = 0)

4. **Purcell 1.46: 3 charged beads constrained to move on a circular wire
Helpful fact: at equilibrium, the component of the net electric force on the bead TANGENT TO THE CIRCLE must be zero. Any nonzero RADIAL component of the net electric force on the bead will be balanced by the normal force exerted by the wire on the bead.
NOTE: to find the position of the third charge, use symmetry to make the tangential component of the force zero.
When you set up the equations to find the magnitude of the third charge, find all distances in terms of the radius of the circle using trigonometry (the Law of Cosines can also be used c2 = a2+b2-2 a b cos(gamma))

PROBLEM SOLVING CHECKLIST: to be checked for EACH AND EVERY PROBLEM you hand in!

0. NEVER write only the answer. You need to show where it came from and why it is right.


1. Check that you have drawn at least one diagram and labelled it clearly. This might involve copying the diagram from the problem statement and adding extra labels, or making a diagram from scratch. Make it pretty large to leave room for labels and make it readable. Quantities from the statement of the problem should appear in the diagram.

2. Check that the meanings of any symbols you have introduced yourself are clear. First, if the symbol is a label in the diagram, make sure the diagram makes its meaning clear. If not, or if it is not in the diagram, write a short phrase to explain what the symbol represents.

3. Check that all vector quantities are written so that their vector character is clear. In typesetting, boldface indicates a vector quantity. In a diagram, if you draw an arrow to indicate the direction and write the magnitude next to the arrow (this is how we draw force diagrams). In handwriting, put an arrow over the quantity to show it is a vector. Velocity is a vector, so write v with an arrow over it. The x component of the velocity is not a vector, so do not put an arrow over it. In a sum, you cannot add a vector to a number. In an equation, you cannot have a vector on one side and a number on the other side.

4. Check that each statement clearly follows from the previous statement. (This is in the same spirit as Checklist Item #0 - you can't state things without showing where they came from).

5. Check that you have stated the answer at the end of the solution. The answer should be an equation in the form
desired quantity = expression in terms of given quantities
You might be tempted to save writing by just writing the right hand side, but it is best practice to write the whole thing (this acts as a final check that your expression is indeed for the quantity the problem asked for).
To make it clear that it is the answer, PUT A BOX AROUND IT.

SOLVING PROBLEMS INVOLVING SMALL OSCILLATIONS

All small oscillation problems should be approached the same way.
(0) Draw a diagram of the equilibrium state and identify the "small displacement" quantity (it might be an angle).
Then, if the forces in the problem are conservative, you have a choice:
(1) you can compute the restoring force (or torque if the small displacement is an angle) and put it into a form proportional to the small displacement quantity (this might involve a linear approximation to a function of the quantity) OR
(1') you can write the kinetic and potential energies, and expand the potential energy to QUADRATIC order in the small displacement quantity).
Finally, depending on your choice above, do
(2) In the force/torque case, then look at your Newton's law/torque equation and map the quantities to the spring equation m x.. = - k x to get the frequency f = root (k/m)/(2 pi)., OR
(2') In the energy case, look at the mechanical energy and map the quantities to the spring mechanical energy (1/2) m x.^2 + (1/2) k x^2 to get the frequency f = root (k/m)/(2 pi).
If there are nonconservative forces (eg kinetic friction) in the problem then you HAVE TO compute the restoring force or torque
(steps (1) and (2)).

ANSWERS
1. 2.9 x 10-6 C
2. ((4kqQ/(ml3))^(1/2))/(2 pi)
3. proof
4. q Sqrt[2] (cos pi/8)2/(sin pi/8)