object objectspring
m1
x10 x2
m2
position
coordinates
MATH 164, Spring 2001 Due Date: Name(s):
Honors Project 5: Coupled Spring-Mass Systems
In Honors Project 4 we considered spring-mass systems in which a single object was connected to a ground
by a single spring. There are numerous variations on such systems, and these variations are important in
applications. In this project we consider two such variations.
We begin with the situation in which two objects of mass m1
and m2 are connected by a single spring whose spring constant is
k and whose unstretched equilibrium length is L. We assume the
objects move horizontally with no friction or damping due to the
environment.
Applying Newton's Second Law of Motion to each object, we find that
m1
d2
x1
dt2
= F1 and m2
d2
x2
dt2
= F2
where Fi is the force acting on object i, i = 1, 2. Since the only force acting on each object is (according to
Hooke's law) proportional to the distance the spring is stretched out of equilibrium, it follows that
m1
d2
x1
dt2
= k(x2 - x1 - L) and m2
d2
x2
dt2
= -k(x2 - x1 - L).
We now make two observations:
First,
m1
d2
x1
dt2
+ m2
d2
x2
dt2
=
d2
dt2
(m1x1 + m2x2) = 0,
so the center of mass of our system
m1x1 + m2x2
m1 + m2
does not accelerate: it moves at constant velocity (and hence obeys Newton's First Law of Motion).
Second, if z = x2 - x1 - L then it follows that
d2
z
dt2
= -k
1
m1
+
1
m2
z ()
so the stretching of the spring exhibits simple harmonic motion. Indeed, if we let
1
m
=
1
m1
+
1
m2
,
so the reduced mass
m =
m1m2
m1 + m2
then
d2
z
dt2
= -
k
m
z.
Problems:
1. Prove equation (*).
object
m
2. Consider the situation in which an object of mass m is connected to one wall by a spring whose spring
constant is k1 and whose unstretched equilibrium length is L1 and to an opposite wall by a spring
whose spring constant is k2 and whose unstretched equilibrium length is L2. (See the figure below.)
Observe that if D is the distance between the walls, then D is not necessarily equal to L1 + L2. (Can
you see why?) We finally assume the objects move horizontally with no friction or damping due to the
environment. What can you say about the motion of the object? Specifically, does it illustrate simple
harmonic motion? Explain.