Motion of a Particle Attached to Two Springs on a Smooth Surface
A particle between two springs on a table can execute simple harmonic motion if pulled to one side.
A particle of mass m is attached to two strings of natural lengths
and
and force constants
and
respectively. The ends of the springs are attached to endpoints A and B, so that
and both springs are initially in tension. Suppose that the mass is pulled to the right from its equilibrium position so that the length of spring 1 is
and the length of spring 2 is
The tension in spring 1 is
and the tension in spring 2 is
The resultant force on the mass is
Using
gives
(1)
When the system is in equilibrium,
so that
Let
then
and
Substitute this into (1) to give
Hence
and the motion is simple harmonic with period