Download Take Home Assignment that Counts as a test grade GREMLIN FREE

Take Home Assignment that Counts as a test grade
GREMLIN FREE ZONE!
Name___________________
Hour____
Score____/135 pts
1. An amusement park ride features a passenger
compartment of mass M that is released from rest
at point A, as shown in the figure above, and
moves along a track to point E. The compartment
is in free fall between points A and B, which are a
distance of 3R/4 apart, then moves along the
circular arc of radius R between points B and D.
Assume the track is frictionless from point A to
point D and the dimensions of the passenger
compartment are negligible compared to R.
(a) On the dot below that represents the passenger compartment, draw and label the forces (not
components) that act on the passenger compartment when it is at point C, which is at an angle  from
point B. (Hint: two forces) (2pts)
(b) In terms of  and the magnitudes of the forces drawn in part (a), determine an expression for the
magnitude of the centripetal force acting on the compartment at point C. If you need to draw anything
besides what you have shown in part (a) to assist in your solution, use the space below. Do NOT add
anything to the figure in part (a). (Hint: Newton’s 2nd Law, split up weight!) (2pts)
(c) Derive an expression for the speed v of the passenger compartment as it reaches point D in terms of
M, R, and fundamental constants, as appropriate. (hint: use energy…. The normal force varies direction
and magnitude making the acceleration NOT constant, that means your four equations are NOT usable)
(5pts)
A force acts on the compartment between points D and E and brings it to rest at point E.
(d) If the compartment is brought to rest by friction, calculate the numerical value of the coefficient of
friction  between the compartment and the track. (hint: multiple possible approaches here…. Forces or
work or conservation of energy…. Hockey puck slowing down on ice!! No GREMLINS) (5pts)
(e) Now consider the case in which there is no friction between the compartment and the track, but
instead the compartment is brought to rest by a braking force -kv , where k is a constant and v is the
velocity of the compartment. Express all algebraic answers to the following in terms of M, R, k, vD , and
fundamental constants, as appropriate.
i. Write, but do NOT solve, the differential equation for v(t) . (Newton’s 2nd Law with one substitution)
(2pts)
ii.
\
Solve the differential equation you wrote in part i. (look at Capacitors discharging in notes) (5pts)
iii.
On the axes below, sketch a graph of the magnitude of the acceleration of the compartment as a
function of time. On the axes, explicitly label any intercepts, asymptotes, maxima, or minima with
numerical values or algebraic expressions, as appropriate. (magnitude means ignore that the
acceleration is negative the whole time…..acceleration is NOT constant… this will be a curve) (2pts)
2. A spherical cloud of charge of radius
R contains a total charge +Q with a
nonuniform volume charge density
that varies according to the equation
where r is the distance from the center of
the cloud. Express all algebraic answers in
terms of Q, R, and fundamental constants
(a) Determine the following as a function of r
for r > R.
i. The magnitude E of the electric field (Hint: you are outside… it is all enclosed!) (5pts)
ii. The electric potential V (5pts)
(b) A proton is placed at point P shown above and released. Describe its motion for a long time
after its release. (Hint: include a description of velocity and acceleration…. Complete sentences
NOT required) (5pts)
(c) An electron of charge magnitude e is now placed at point P. which is a distance r from
the center of the sphere, and released. Determine the kinetic energy of the electron as a function
of r as it strikes the cloud. (Hint: use energy because the force depends on distance and is not
constant) (5pts)
(d) Derive an expression for ρo (Hint: integrate to find total charge with limits 0 to R….
MULTIPY before you do power rule!!!!.... this problem might be in your notes!) (5pts)
(e) Determine the magnitude E of the electric field as a function of r for r < R (Hint: you have
to integrate first to find charge enclosed then you can do Gauss’ Law) (5pts)
3. A student performs an experiment to determine the emf
and internal resistance r of a given battery. The student
connects the battery in series to a variable resistance R,
with a voltmeter across the variable resistor, as shown
in the figure to the right, and measures the voltmeter
reading V as a function of the resistance R. The data are
shown in the table below.
i. Derive an expression for the measured voltage V. Express your answer in terms of R, ε, r,
and physical constants, as appropriate. (Hint: Loop rules are your friend!) (5pts)
ii. Rewrite your expression from part (a)-i express 1/V as a function of 1/R. (3 pts)
(b) On the grid below, plot data points for the graph of 1/V as a function of 1/R. Clearly scale
and label all axes, including units as appropriate. Draw a straight line that best represents the
data. (5pts)
c. use the straight line from part (b) to obtain values for the following. (10pts)
i.
ε
ii. r
(d) using the results of the experiment, calculate the maximum current that the battery can
provide. (5pts)
(e) A voltmeter is to be used to determine the emf of the battery after removing the battery
from the circuit. Two voltmeters are available to take this measurement- one with low internal
resistance and one with high internal resistance. Indicate which voltmeter will provide the
most accurate measurement.
___The voltmeter with low resistance will provide the most accurate measurement
___The voltmeter with high resistance will provide the most accurate measurement
___the two voltmeters will provide equal accuracy
Justify your answer (Hint: This ap problem was from 2015… perhaps you could check your
own answer, while there maybe you could practice some more problems! ) (5pts)
4. The circuit above contains a capacitor of
capacitance C, a power supply of emf ε,
two resistors of resistances R1, and R2,
and two switches. S1 and S2 . Initially, the
capacitor is uncharged and both switches
are open. Switch S 1, then gets closed at
time t = 0.
(a)
Write a differential equation that can be
solved to obtain the charge on the capacitor as a
function of time t. (hint: Loop rule)(2pt)
(b)
Solve the differential equation in part (a) to determine the charge on the capacitor as a
function of time t. Numerical values for the components are given as follows: (Hint: Capacitor
charging)(5pts)
ε=10V
C = 0.05 F
R1 = R2 =4600 Ω
(c)
Determine the time at which the capacitor has a voltage 4 V across it. (use that
equation from part b) (5pts)
After switch S1 has been closed for a long time, switch S2, gets closed at a new time t = 0.
(d) On the axes below, sketch graphs of the current I1, in R1, versus time and of the current
I2, in R2, versus time, beginning when switch S2, is closed at new time t = 0. Clearly label
which graph is I1, and which is I2. (3pts)
5. A section of a long conducting cylinder with inner radius a and outer radius b carries a current I0
that has a uniform current density, as shown in the figure above.
(a) Using Ampère’s law, derive an expression for the magnitude of the magnetic field in the following
regions as a function of the distance r from the central axis. (Hint: See notes)
i. r < a (5pts)
____________________________________________________________________________
ii. a < r < b (5pts)
________________________________________________________________________
iii. r = 2b (5pts)
(b) On the cross-sectional view in the diagram above, indicate the direction of the field at point P, which
is at a distance r = 2b from the axis of the cylinder. (Hint: previous page…. Draw an arrow) (2pts)
(c) An electron is at rest at point P. Describe any electromagnetic forces acting on the electron. Justify
your answer. (2pts)
Now consider a long, solid conducting cylinder of radius b carrying a current I0 .
(d)
Use Ampere’s Law to derive an expression for the magnitude of the magnetic field inside
this cylinder as a function of r .(5pts)
An experiment is conducted using a particular solid cylinder of radius 0.010 m carrying a current of 25 A.
The magnetic field inside the cylinder is measured as a function of r, and the data is tabulated below.
(e)
i. On the graph below, plot the data points for the magnetic field B as a function of the distance r, and
label the scale on both axes. Draw a straight line that best represents the data. (5pts)
ii. Use the slope of your line to estimate a value of the permeability µ0 . (5pts…. Show work on how you
determine slope!!!)