2.6.3. Suppose a string is stretched with tension + horizontally between two anchors
at x = 0 and x = 1. At each of the n - 1 equally spaced positions xx = k/n,
k = 1,...,n-1, we attach a little mass m; and allow the string to come to equi-
librium. This causes vertical displacement of the string. Let q, be the amount
of displacement at x. If the displacements are not too large, then an approxi-
mate force balance equation is
nT(9k - 9k-1)+ nT(9k - 9k+1)= meg,
k =1,...,n-1,
where g =-9.8 m/s is the acceleration due to gravity, and we naturally define
90 = 0 and 97 = 0 due to the anchors.
(a) Show that the force balance equations can be written as a linear system
Aq = f, where q is a vector of displacements and A is a tridiagonal matrix
(that is, A;; = 0 if |i - j| > 1) of size (n - 1) x (n-1).
(b) Let T = 10 N, and let my = (1/10n) kg for every k. Find the displace-
ments in MATLAB when n = 4 and n = 40, and superimpose plots of q
over 0 < x _ 1 for the two cases. (Be sure to include the zero values at
x = 0 and x = 1 in your plots of the string.)
(c)
Repeat (b) for the case my = (k/5n?) kg.