Download Homework due 5-7

Homework due 5-7-2013
1. A uniform chain of mass M and length l, hangs from a hook in
the ceiling. The bottom link is now raised vertically and hung on the
hook as shown above on the right.
a. Determine the increase in gravitational potential energy of the
chain by considering the change in position of the center of mass
of the chain
b Write an equation for the upward external force F(y) required to
lift the chain slowly as a function of the vertical distance y,
c. Find the work done on the chain by direct integration of
2. A positive charge distribution exists within a nonconducting spherical region of radius a.
The volume charge density p is not uniform but varies with the distance r from the center of
the spherical charge distribution, according to the relationship ρ = βr for 0 < r < a, where β is
a positive constant, and ρ=0, for r >a.
a. Show that the total charge Q in the spherical region of radius a is βπa 4
b. In terms of β, r, a, and fundamental constants, determine the magnitude of the electric field at
a point a distance r from the center of the spherical charge distribution for each of the
following cases. i. r> a
ii. r =a
iii. 0 < r <a
c. In terms of β, a, and fundamental constants, determine the electric potential at a point a
distance r from the center of the spherical charge distribution for each of the following cases
i. r= a
ii. r =0
3. A skier of mass .M is skiing down a frictionless hill that makes an angle θ with the
horizontal, as shown in the diagram.
The skier starts from rest at time t =0
and is subject to a velocity-dependent
drag force due to air resistance of the
form F= -bv, where v is the velocity of
the skier and b is a positive constant.
Express all algebraic answers in terms of M, b, θ, and fundamental constants.
(a). On the dot below that represents the skier, draw a free-body
diagram indicating and labeling all of the forces that act on the skier
while the skier descends the hill.
(b) Write a differential equation that can be used to solve for the velocity
of the skier as a function of time.
(c) Determine an expression for the terminal velocity vT, of the skier.
(d) Solve the differential equation in part (b) to determine the velocity of the skier as a function
of time. Showing all your steps.
Homework due 5-7-2013
(e) On the axes below, sketch a graph of the acceleration a of the skier as a function of time
t, and indicate the initial value of a. Take downhill as positive.
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.
(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:
ε=12V
C = 0.060 F
R1 = R2 =4700 Ω
(c)
Determine the time at which the capacitor has a voltage 4.0 V across it. 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.