Applied Mathematics I (500.303)

1. Kreyszig problem 1.1.22
2. Kreyszig problem 1.1.24
3. Kreyszig problem 1.4.14
4. Kreyszig problem 1.4.16
5. Kreyszig problem 1.6.42
6. Kreyszig problem 1.6.43
7. Kreyszig problem 2.3.13
8. Kreyszig problem 2.3.17
9. Kreyszig problem 2.3.19

   10.   The following problem is based on the material in Section 1.5.

Suppose that a flexible 4-ft rope starts with 3 ft of its length arranged in a heap at the right edge of a high horizontal table, with the remaining foot hanging (at rest) off the table.   At time t = 0 the heap begins to unwind and the rope begins gradually to fall off the table, under the force of gravity pulling on the overhanging part.   We make the following assumptions:

• Frictional forces of all sorts are negligible.
• Some unspecified device prevents the entire heap from simply falling off the edge all at once.

We want to determine how long it will take for all the rope to fall of the table.   Let

• x(t) denote the length of overhanging rope at any time t
• v(t) denote its velocity with the positive direction downward

(a)   What are the initial conditions (at time t = 0) for this problem?

(b)   The gravitational force acting upon the rope is due solely to its weight, i.e. F = mg.   If the linear density of the rope is w (slugs/foot), what is the gravitational force acting upon the rope?

(c)   Note that the mass of the overhanging rope varies with time.    Thus, Newton's second law takes the form F = d/dt(mv).   Show that F = d/dt(mv) = w(x dv/dt + v dx/dt).

(d) Use the answers to parts (b) and (c) to derive the equation (v²/x - g) dx + v dv = 0.

(e)   Show that the equation in (d) is NOT exact.   Then, find an integrating factor that's a function of x alone that will make the equation exact.   Show that the new equation (i.e., the one resulting after you apply the integrating factor) is exact.

(f)   Find the general solution for new equation you got in (f).   Use the initial conditions to determine the value of the integration constant.

Note:   In order to actually determine the time t when all the rope has fallen to the floor requires manipulation of an improper integral.