082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

19.6 Constant coefficient equations 565
Engineeringapplication19.6
Oscillatingmass--springsystem
Consider the simple mechanical problem of a mass resting on a smooth frictionless
table. A spring is attached to the mass and an adjacent anchor point as shown in
Figure 19.6. The spring is capable of being compressed as well as stretched. When
the spring isneither compressed nor stretched the mass islocated atx = 0.
m
0
x
Figure19.6
Mass--spring system.
If the mass is pulled in thexdirection and let go, it will oscillate about thex = 0
position. We wish to find the position of the mass,x, as a function of time,t. Differential
equations are needed todescribe thisproblem fully.
Newton’s second law states that if a forceF is applied to a body of massmthen
themotionofthebodyisgovernedbyF =ma,whereaistheacceleration.Applying
Newton’s second law, and noting thata = d2 x
dt2,we obtain
F =m d2 x
dt 2
The force, F, is provided by the spring. The force exerted by a spring is given by
Hooke’slaw,whichstatesthattheforceisproportionaltotheextensionorcompression
of the spring,
F=−kx
wherekisthespringconstantforthespringinuse.Theminussignisrequiredsothat
when the spring is stretched (x > 0) the force is in the negativexdirection. When
the spring iscompressed (x < 0) the force isinthe positivexdirection.
If the table is sufficiently smooth and if there are no other external forces acting,
thenm d2 x
dt 2 = −kx.
Therefore thedifferentialequationthatgoverns motioninthe system is
m d2 x
dt 2 +kx=0
We write thisas
d 2 x
dt 2 + k m x = 0
Let k m = ω2 ,giving
d 2 x
dt 2 +ω2 x=0
➔