Download Homework due 5-8

Homework Due 5-8-2013
1.
A stick of length 2L and negligible
mass has a point mass m affixed to
each end. The stick is arranged so that
it pivots in a horizontal plane about a
frictionless vertical axis through its
center. A spring of force constant k is
connected to one of the masses as
shown above. The system is in
equilibrium when the spring and stick
are perpendicular. The stick is displaced through a small angle θ, as shown and then
released from rest at t = 0
a. Determine the restoring torque when the stick is displaced from equilibrium through the small
angle θ,
b. Determine the magnitude of the angular acceleration of the stick just after it has been
released.
c. Write the differential equation whose solution gives the behavior of the system after it
has been released.
d. Write the expression for the angular displacement θ of the stick as a function of time t
after it has been released from rest
2.
A ferryboat of mass M1= 2.0 x
105 kilograms moves toward a
docking bumper of mass M 2 that
is attached to a shock absorber.
Shown below is a speed v vs. time
t graph of the ferryboat from the
time it cuts off its engines to the
time it first comes to rest after colliding with the bumper. At the instant it hits the
bumper, t = 0 and v = 3 meters per second
After colliding inelastically with the bumper, the
ferryboat and bumper move together with an initial
speed of 2 meters per second. Calculate the mass of
the bumper M2.
a. After colliding, the ferryboat and bumper move
with a speed given by the expression v = 2e -4t.
Although the boat never comes precisely to rest, it
travels only a finite distance. Calculate that distance.
b. While the ferryboat was being slowed by water
resistance before hitting the bumper, its speed was given by 1/v = 1/3 + βt, where β is a
constant. Find an expression for the retarding force of the water on the boat as a function of
speed
Homework Due 5-8-2013
3. The figure above
left shows a hollow,
infinite, cylindrical,
uncharged conducting
shell of inner radius r1,
and outer radius r2 .
An infinite line
charge of linear
charge density +λ is
parallel to its axis but
off center. An
enlarged cross section
of the cylindrical shell
is shown above right.
(a) On the cross
section above right,
i.
sketch the electric field lines, if any, in each of regions I, II, and III and
ii. use + and - signs to indicate any charge induced on the conductor.
(b) In the spaces below, rank the electric potentials at points a, b, c, d, and e from highest
to lowest (I = highest potential). If two points are at the same potential, give them the same
number.
____Va
____Vb
_____Vc
_____Vd
_____Ve
(c) The shell is replaced
by another cylindrical
shell that has the same
dimensions but is
nonconducting and
carries a uniform volume
charge density +ρ. The
infinite line charge, still
of charge density +λ is
located at the center of
the shell as shown above.
Using Gauss's law,
calculate the magnitude
of the electric field as a
function of the distance r from the center of the shell for each of the following regions.
Express your answers in terms of the given quantities and fundamental constants.
i.
r < r1
ii. r1 < r < r2
iii. r>r2