Download 1977- Electricity and Magnetism I A charge +Q is uniformly

1977- Electricity and Magnetism I
A charge +Q is uniformly distributed around
a wire ring of radius R. Assume that the
electric potential is zero at x = infinity, with
the origin 0 of the x-axis at the center of the
ring.
a. What is the electric potential at a point P on the x-axis?
b. Where along the x-axis is the electric potential the greatest? Justify your answer.
c. What is the magnitude and direction of the electric field E at
point P?
d. On the axes below, make a sketch of E as a function of the
distance along the x-axis showing significant features
e. Find the electric field at point P if this was a solid disk of charge
+Q
1989- Electricity and Magnetism III
A battery with an emf of 20 volts is connected in
series with a resistor of 300,000 ohms and an
air-filled parallel-plate capacitor of capacitance
6 microfarads.
a. Determine the energy stored in the capacitor
when it is fully charged.
The spacing between the capacitor plates is suddenly increased in a time short compared to the time
constant of the circuit to four times its original value.
b. Determine the work that must be done in increasing the spacing in this fashion.
c. Determine the current in the resistor immediately after the spacing is increased.
After a long time, the circuit reaches a new static state.
d. Determine the total charge that has passed through the battery.
e. Determine the energy that has been added to the battery
1990- Electricity and Magnetism I
A sphere of radius R is surrounded by a concentric spherical
shell of inner radius 2R and outer radius 3R, as shown above.
The inner sphere is an insulator containing a net charge + Q
distributed uniformly throughout its volume. The spherical shell
is a conductor containing a net charge + q different from + Q.
Use Gauss's law to determine the electric field for the following
values of r, the distance from the center of the insulator.
a. 0 < r < R
b. R < r < 2R
c. 2R < r < 3R
Determine the surface charge density (charge per unit area) on
d. the inside surface of the conducting shell,
e. the outside surface of the conducting shell
2003 Electricity and Magnetism III
An airplane has an aluminum antenna attached to its wing that extends 15 in from wingtip to wingtip. The
plane is traveling north at 75 m/s in a region where Earth's magnetic field has both a vertical component
and a northward component, as shown above. The net
magnetic field is at an angle of 55 degrees from
horizontal and has a magnitude of 6.0 x 10-5 T.
(a)
On the figure below, indicate the direction of
the magnetic force on electrons in the antenna. Justify
your answer.
(b) Determine the magnitude of the electric field generated
in
the antenna.
(c) Determine the potential difference between the ends of
the antenna.
(d) On the figure below, indicate which end of the antenna is at
higher potential
(e) The ends of the antenna are now connected by a conducting
wire so that a closed circuit is formed.
i. Describe the condition(s) that would be necessary for a
current to be induced in the circuit. Give a specific example of
how the condition(s) could be created.
ii. For the example you
gave in i. above, indicate
the direction of the
current in the antenna on
the figure below
1996- Mechanics I
A thin, flexible metal plate
attached at one end to a
platform, as shown above,
can be used to measure
mass. When the free end
of the plate is pulled down
and released, it vibrates in
simple harmonic motion
with a period that depends
on the mass attached to the
plate. To calibrate the force constant, objects of known mass are attached to the plate and the plate is
vibrated, obtaining the data shown below.
a. Fill in the blanks in the data table.
b. On the
graph below, plot T versus
mass. Draw on the graph the
line that is your estimate of
the best straight-line fit to
the data points.
c. An object whose mass is
not known is vibrated on
the plate, and the average
time for ten vibrations is
measured to be 16.1 s. From
your graph, determine the
mass of the object. Write
your answer with a reasonable number of significant digits.
d. Explain how one could determine the force constant of the metal plate.
e. Can this device be used to measure mass aboard the space shuttle Columbia as it orbits the Earth?
Explain briefly.
f If Columbia is orbiting at 0.3 x 106 m above the Earth's surface, what is the acceleration of Columbia
due to the Earth's gravity? (Radius of Earth = 6.4 x 6 m, mass of Earth = 6.0 x 1024 kg)
g. Since the answer to part (f) is not zero, briefly explain why objects aboard the orbiting Columbia seem
weightless
1986- Mechanics III
A special spring is constructed in which the restoring force is in the opposite direction to the
displacement, but is proportional to the cube of the displacement; i.e.,
F = -kx3
This spring is placed on a horizontal frictionless surface. One end of the spring is fixed, and the
other end is fastened to a mass M. The mass is moved so that the spring is stretched a distance A and
then released. Determine each of the following in terms of k, A, and M
a. The potential energy in the spring at the instant the mass is released
b. The maximum speed of the mass
c. The displacement of the mass at the point where the potential energy of the spring and the
kinetic energy of the mass are equal
The amplitude of the oscillation is now increased
d. State whether the period of the oscillation increases, decreases, or remains the same. Justify your
answer
1978 Mechanics III
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
1981 Mechanics III
A thin, uniform rod of mass M, and length L is
initially at rest on a frictionless horizontal surface.
The moment of inertia of the rod about its center of
mass is ML2/12. As shown in Figure I, the rod is
struck at point P by a mass m 2 whose initial
velocity v is perpendicular to the rod. After the
collision, mass m2, has velocity - ½ v as shown in
Figure II. Answer the following in terms of the
symbols given.
a. Using the principle of conservation of linear momentum, determine the velocity v' of the center
of mass of this rod after the collision.
b. Using the principle of conservation of angular momentum, determine the angular velocity ω of the
rod about its center of mass after the collision.
c. Determine the change in kinetic energy of the system resulting from the collision.