point mass point massharmonic
spring
m1
x1 x2
m2
position
coordinates
MATH 261, Spring 2001 Due Date: Name(s):
Honors Project 13: Mechanical Energy of a Coupled Spring-Mass System
Quantities associated with a physical system can sometimes be simplified by changing the frame of reference.
For example:
The total mechanical energy H of the coupled spring-mass system illustrated below is given by
H(x1, x2, p1, p2) =
p2
1
2m1
+
p2
2
2m2
+
1
2
(x2 - x1 - x0)2
where x0 is the equilibrium (unstretched) length of the spring, and pi = mivi where vi = dxi/dt, for i = 1, 2.
Problem: Assuming we have defined two new coordinates
r  x2 - x1 - x0 and s 
m1x1 + m2x2
m1 + m2
and that
pr 
m1m2
m1 + m2
vr and ps  (m1 + m2)vs,
where vr = dr/dt and vs = ds/dt, express H as a function of r, s, pr, and ps in a simplified form.
Contributor: Dr. Clifford Dykstra
Department of Chemistry, IUPUI