Physics 272: Advanced Honors Physics II
due in class Monday, January 22, 2018
Reading: Purcell 1.1-1.5
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
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,
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+x2+... (this one is used in
the energy approach)
= 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
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))
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
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.
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
(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
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)).
x 10-6 C
2. ((4kqQ/(ml3))^(1/2))/(2 pi)
4. q Sqrt (cos pi/8)2/(sin pi/8)