The below sketch shows a moveable platform (blue), attached to a spring s1 and free to
slide over a frictionless tabletop. On the platform is a block (green) connected to the
platform by a spring s2. The contact between block and platform is also frictionless. The
equilibrium position of the right end of s1 is indicated by a marker (red) fixed to the
tabletop (and displaced laterally to allow the platform to oscillate without striking the
marker). The equilibrium position of the right end of s2 is indicated by a similar marker
fixed to the platform. When the platform and block are in equilibrium, the displacement
in the x-direction of both markers and of the left edge of the block coincide.
We are interested in the motion of the block with respect to the table when both the
platform and the block are oscillating. That is, we are interested in the net displacement
of the (left edge) of the block from the marker that is fixed to the tabletop. Call this
displacement x.
A. Express x in terms of x1 (the displacement of the platform-fixed marker from the
table-fixed marker) and x2 (the displacement of the block from the platform-fixed
marker).
B. Suppose that the force constants of the springs and masses of platform and block
have been chosen so that the period of the platform’s oscillation over the tabletop is
exactly equal to the period of the block’s oscillation over the platform.
1. Write the most general expression for x1 as a function of time.
2. Write the most general expression for x2 as a function of time.
3. Write the most general expression for x as a function of time.
Some of the constants you used in part B1 may have the same values as constants you
used in part B2. Which constants necessarily have the same value and which can be
different?
C. Let  denote the phase difference between the platform’s oscillation over the
tabletop and the block’s oscillation over the platform. Describe in words the
block’s displacement over the tabletop when the amplitudes of the two
oscillations are equal and when
i.  = 0
ii.  = 
iii.  = /2
iv  = -/2