Download lesson homework Tuesday may 1st

2010 Mechanics III
A skier of mass m will be pulled up a hill by a rope, as shown above. The of
magnitude
the magnitude
of the acceleration
as a function of time t can be modeled by the equations
where amax and T are constants. The hill is inclined at an angle Ө above the horizontal, and friction
between the skis and the snow is negligible. Express your answers in terms of given quantities and
fundamental constants. negligible.
(a) Derive an expression for the velocity of the skier as a function of time during the acceleration.
Assume the skier starts from rest.
(b) Derive an expression for the work done by the net force on the skier from rest until terminal speed is
reached.
(c) Determine the magnitude of the force exerted by the rope on the skier at terminal speed.
(d) Derive an expression for the total impulse imparted to the skier during the acceleration.
(e) Suppose that the magnitude of the acceleration is instead modeled as a = amaxe-πt/2T for all t > 0 ,
where amax and T are the same as in the original model. On the axes below, sketch the graphs of the
force exerted by the rope on the skier for the two models., from t = 0 to a time t > T . Label the
original model F1 and the new model F2.
2003 Electricity and Magnetism I
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
ii. The electric potential V
(b) A proton is placed at point P shown above and released. Describe its motion for a long time after its
release.
(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.
(d) Derive an expression for ρo
(e) Determine the magnitude E of the electric field as a function of r for r < R
2002- Mechanics I
A crash test car of mass 1,000 kg moving at constant speed of
12 m/s collides completely inelastically with an object of
mass M at time t = 0. The object was initially at rest. The
speed v in m/s of the car-object system after the collision is
given as a function of time t in seconds by the expression
(a) Calculate the mass M of the object.
(b) Assurning an initial position of x = 0, determine an expression for the position of the car-object
system after the collision as a function of time t.
(c) Determine an expression for the resisting force on the car-object system after the collision as a
function of time t.
(d) Determine the impulse delivered to the car-object system from t = 0 to t = 2.0 s
Homework Tuesday
A ferryboat of mass M 1= 2.0 x 10 5
kilograms moves toward a docking bumper
of mass M2 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
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
A thin plastic rod has uniform linear positive-charge density λ. The rod is
bent into a semicircle of radius R as shown above.
a. Determine the electric potential Vo at point 0, the center of the semicircle.
b. Indicate on the diagram above the direction of the electric field at point O.
Explain your reasoning.
c. Calculate the magnitude Eo of the electric field at point O.
d.
Write an approximate expression, in terms of q, V ∞ and Eo, for the
work required to bring a positive point charge q from infinity to point P,
located a small distance s from point O as shown in the diagram above.