PROBLEM 1
A metal ball is buried in an ice cube that is in a bucket of water.
(a) If the ice cube with the metal ball is initially under water, what
... [Show More] happens to the water level
when the ice melts?
(b) If the ice cube with the metal ball is initially floating in the water, what happens to the water
level when the ice melts?
(c) Explain how the Earth’s average sea level could have increased by at least 100 m compared
to about 20,000 years ago.
(d) Estimate the thickness and weight per unit area of the ice that melted in (c). You may wish
to use the fact that the density of ice is 920 kg m-3, today the land surface area of the Earth is about
148,300,000 km2 and water area is about 361,800,000 km2.
PROBLEM 2
Sketch and find the volume of the largest and smallest convex plug manufactured from a sphere
of radius r = 1 cm to fit exactly into a circular hole of radius r = 1 cm, an isosceles triangle with
base 2 cm and a height h = 1 cm, and a half circle radius r = 1 cm and base 2 cm.
PROBLEM 3
An initially stationary particle mass m1 is on a frictionless table surface and another particle
mass m2 is positioned vertically below the edge of the table. The distance from the particle mass
m1 to the edge of the table is l. The two particles are connected by a taught, light, inextensible
string of length L > l.
(a) How much time elapses before the particle mass m1 is launched off the edge of the table?
(b) What is the subsequent motion of the particles?
(c) How is your answer for (a) and (b) modified if the string has spring constant k?
PROBLEM 4
The velocity of water waves in shallow water may be approximated as where g is the
acceleration due to gravity and h is the depth of the water. Sketch the lowest frequency standing
water wave in a 5 m long garden pond that is 0.9 m deep and estimate its frequency.
PROBLEM 5
(a) What is the dispersion relation of a wave whose group velocity is half the phase velocity?
(b) What is the dispersion relation of a wave whose group velocity is twice the phase velocity?
(c) What is the dispersion relation when the group velocity is four times the phase velocity?
PROBLEM 6
A stationary ground-based radar uses a continuous electromagnetic wave at 10 GHz frequency
to measure the speed of a passing airplane moving at a constant altitude and in a straight line at
1000 km hr-1. What is the maximum beat frequency between the out going and reflected radar
beams? Sketch how the beat frequency varies as a function of time. What happens to the beat frequency
if the airplane moves in an arc?
v = gh
Applied quantum mechanics 3
PROBLEM 7
How would Maxwell’s equations be modified if magnetic charge g (magnetic monopoles) were
discovered? Derive an expression for conservation of magnetic current and write down a generalized
Lorentz force law that includes magnetic charge. Write Maxwell’s equations with magnetic
charge in terms of a field .
PROBLEM 8
The capacitance of a small metal sphere in air is . A thin dielectric film
with relative permittivity uniformly coats the sphere and the capacitance increases to
. What is the thickness of the dielectric film and what is the single electron charging
energy of the dielectric coated metal sphere?
PROBLEM 9
(a) A diatomic molecule has atoms with mass m1 and m2. An isotopic form of the molecule has
atoms with mass m'1 and m'2. Find the ratio of vibration oscillation frequency w / w' of the two
molecules.
(b) What is the ratio of vibrational frequencies for carbon monoxide isotope 12 ( ) and
carbon monoxide isotope 13 ( )?
PROBLEM 10
(a) Find the frequency of oscillation of the particle of mass m illustrated in the Fig. The particle
is only free to move along a line and is attached to a light spring whose other end is fixed at point A
located a distance l perpendicular to the line. A force F0 is required to extend the spring to length l.
(b) Part (a) describes a new type of child’s swing. If the child weighs 20 kg, the length l = 2.5
m, and the force F0 = 450 N, what is the period of oscillation? [Show Less]