Homework Due 4-30-2013
1. Students are to conduct an experiment to investigate the relationship between the terminal speed of a stack of falling paper coffee filters and its mass. Their procedure involves stacking a number of coffee filters, like the one shown in the figure above, and dropping the stack from rest. The students change the number of filters in the stack to vary the mass m while keeping the shape of the stack the same. As a stack of coffee filters falls, there is an air resistance (drag) force acting on the filters.
(a) The students suspect that the drag force FD is proportional to the square of the speed v:
FD = Cv2 , where C is a constant. Using this relationship, derive an expression relating the terminal speed vT to the mass m.
The students conduct the experiment and obtain the following data.
(i) Assuming the functional relationship for the drag force above, use the grid below to plot a linear graph
as a function of m to verify the relationship. Use the empty boxes in the data table, as appropriate,
to record any calculated values you are graphing. Label the vertical axis as appropriate, and place
numbers on both axes.
(ii) Use your graph to calculate C.
A particular stack of filters with mass m is dropped from rest and reaches a speed very close to terminal speed by the time it has fallen a vertical distance Y.
(c)
(i) Sketch an approximate graph of speed versus time from the time the filters are released up to the time
t = T that the filters have fallen the distance Y. Indicate time t = T and terminal speed v =vT on
the graph
(ii) Suppose you had a graph like the one sketched in (c)(i) that had a numerical scale on each axis.
Describe how you could use the graph to approximate the distance Y.
(d) Determine an expression for the approximate amount of mechanical energy dissipated, ΔE , due to air
resistance during the time the stack falls a distance y, where y > Y . Express your answer in terms of y , m,
VT, and fundamental constants.
2. A resistor is made in the form of a cylinder of cross- sectional area A. One portion, of length l1 is made of material whose resistivity is ρ, the other of length l2 is made of material whose resistivity is 3ρ. There is a current I uniformly distributed over the area A. Express all answers in terms of fundamental constants and the symbols shown in the diagram above.
a. Determine expressions for the electric field strengths El and E2 in the two portions of the resistor.
b. Determine the potential difference V between the opposite ends of the resistor.
c. By applying Gauss's law to a surface which encloses the boundary between the two materials,determine the sign and magnitude of the electric charge which is present on this boundary.
3. A spherically symmetric charge distribution has net positive charge Q0 distributed within a radius of R.
Its electric potential V as a function of the distance r from the center of the sphere is given by the following.
for r<R
for r>R
Express all algebraic answers in terms of the given quantities and fundamental constants.
(a) For the following regions, indicate the direction of the electric field E(r) and derive an expression for its magnitude.
i. r < R
____ Radially inward ____ Radially outward
ii. r > R
____ Radially inward ____ Radially outward
(b) For the following regions, derive an expression for the enclosed charge that generates the electric field in that region, expressed as a function of r.
i. r < R
ii. r > R
(c) Is there any charge on the surface of the sphere (r = R) ?
____ Yes ____ No
If there is, determine the charge. In either case, explain your reasoning
(d) On the axes below, sketch a graph of the force that would act on a positive test charge in the regions r < R and r > R. Assume that a force directed radially outward is positive.