Applied Mathematics I (500.303)

1. (Section 2.5)   A mass of 3 kg is attached to the end of a spring that is stretched 20 cm by a force of 15 N.   It is set in motion with initial position x₀ = 0 and initial velocity v₀ = -10 m/s.   (a)   Find the equation governing the motion of the mass.   (b)   Find the amplitude, period, and frequency of the resulting motion.

In problems 2-3, assume that the differential equation of a simple pendulum of length L is

is the gravitational acceleration at the location of the pendulum (at a distance R from the center of the earth; M denotes the mass of the earth).

2. (Section 2.5)   Two pendulums are of lengths L₁ and L₂, and -- when located at the respective distances R₁ and R₂ from the center of the earth -- have periods p₁ and p₂.   Show that

3. (Section 2.5)   A pendulum of length 100.10 in, located at a point at sea level where the distance from the center of the earth is R = 3960 (mi), has the same period as does a pendulum of length 100.00 in atop a nearby mountain.   Use the result of the previous problem to find the height of the mountain.

4. (Section 2.5)   Consider a floating cylindrical buoy with radius r, height h, and density d.   (Recall that the density of water is 1 gm / cm³.)   The buoy is initially suspended at rest with its bottom at the top surface of the water and is released at time t = 0.   Thereafter, it is acted upon by two forces: a downward gravitational force,    F_w ,   equal to its weight and an upward force of buoyancy,   F_b ,   equal to the weight of the water displaced.   At static equilibrium, F_w = F_b.   Let x = x(t) denote the depth of the bottom of the buoy beneath the surface at any time t.
   
   (a) Show that x_e , the point of static equilibrium equals dh.
   
   (b) What is the equation governing the motion of the buoy about x_e ?
   
   (c) Show that the period of motion, equals
   
   (d) Compute the period and amplitude of the motion if d = 0.5 g/cm3 and h = 200 cm.

5. (Section 2.5)   A cylindrical buoy weighing 100 lb floats in water with its axis vertical (as in the previous problem).   When depressed slightly and released, it oscillates up and down four times every 10 seconds.   Assume that friction is negligible.   Find the radius of the buoy.

6. Kreyszig problem 2.9.9
7. Kreyszig problem 2.9.12
8. Kreyszig problem 2.10.5
9. Kreyszig problem 2.11.15