Math 2280-001
Quiz 7 SOLUTIONS
March 6, 2015
1) A focus in this course is a careful analysis of the mathematics and physical phenomena exhibited in
forced and unforced mechanical (or electrical) oscillation problems. Using the mass-spring model, we've
studied the differential equation for functions x t solving
m x##C c x#C k x = F0 cos w t
with m, k, w O 0; c R 0, F0 R 0.
1a) Explain what each of the letters m, k, c, F0 , w represent in this model. Also give their units in the mks
system.
(5 points)
kg$m
Each composite term must have units that reduce to newtons ( 2 ) ...
s
m=mass, units kg
kg
Ns
c=damping coefficient, units
or
s
m
N
kg
k = spring (Hooke's) constant, units
or 2
m
s
radians
w = angular frequency, units
.
s
F0 = forcing amplitude, units N.
s
1b) Describe two of the three forced oscillation phenomena (F0 O 0) that we've studied in this class, i.e.
two of the three phenomena: beating, pure resonance, practical resonance. For the two that you choose
indicate values or relationships between the values, of m, k, w, c lead to this phenomenon. Indicate
briefly the form of the part of the solution formula that reflects this phenomenon. (You don't need to
exhibit the precise solution formula, just the form it takes.)
(5 points)
k
beating: no damping (c = 0), and w z w0 =
, but w s w0 . In this case there is a term in the
m
solution of the form
C cos w t K cos w0 t
with C large. (The other terms are determined by the initial conditions x0 , v0 .)
pure resonance: no damping (c = 0), and w = w0 =
k
. In this case the undetermined coefficients
m
particular solution is of the form
x t = C t sin w0 t
(and the other terms in the solution are determined by x0 , v0 .
practical resonance: relatively small damping c O 0 but c z 0, so that the steady periodic solution
xsp t = C cos w t K a
has relatively large amplitude C when w is close to w0 . ( The precise definition of practical resonance is
that the amplitude C = C w attains its maximum value for some w O 0 as opposed to at w = 0.) (And,
this w will be located approximately at w0 .)