Applications of Linear Equations with Constant Coefficients 277that produces the elongation or compression. (The Englishman Robert Hooke
(1635-1703), a friend oflsaac Newton, first published this result in 1676 as an
anagram and then in detail in 1678. Thus, Hooke discovered Hooke's law at
about the same time calculus was invented.) In the equilibrium position the
only force acting on the system is the force of gravity, so by Hooke's lawmg= k£
where k > 0 is the constant of proportionality for the particular spring. For
a given spring, the spring constant k can easily be calculated by attaching a
body of known mass m to the spring and accurately measuring the resulting
elongation£.T
L Tl
Natural Position L+emFigure 6.2 Mass on a Spring System
We arbitrarily choose the equilibrium position to be the origin and choose
the positive direction to be measured vertically downward from the origin.
See Figure 6.2. It has been shown experimentally that as long as the speed at
which the mass is travelling is not too large, the damping force is proportional
to the speed. The damping force is usually due to the resistance caused by
the medium (air, perhaps) in which the system operates or by the resistance
caused by adding some additional component to the system, such as a dashpot.
Applying Newton's second law of motion to the mass on a spring system with
damping, it can be shown that the position of the mass satisfies the initial
value problem
(4) my" + cy' + ky = O; y(O) = co, y' (0) = c1
where c ;::::: 0 is a constant (the damping constant), k > 0 is the spring constant,
c 0 is the initial location of the mass, and c 1 is the initial velocity of the mass.
If there is an external force j(t) acting on the spring-mass system, such
as a motor which vibrates the support or a magnetic field which acts upon
a suspended iron mass, then the homogeneous initial value problem ( 4) must
be replaced by the nonhomogeneous initial value problem