A particle of mass m is restricted to move without friction on the x-axis. This particle is connected to a fixed point P a distance d from the x-axis by a spring with constant k and length at rest l_0. Working in the two-dimensional plane (x, y), we will denote the coordinates of the particle by (x, 0) while the point P has coordinates (0, d). Find the equilibrium positions of the particle and discuss their stability. In the case where the equilibrium positions are stable, find the period of the small oscillations near equilibrium.

Question

Hint: You will want to distinguish the cases d > l_0, and d < l_0 (ignore the special case d = l_0).

Answer 1

Sorry, I do not know what l_0 means.

Answer 2

To Anonymous: l_0 is the length at rest.

Answer 3

I will call the unstrained length Lo
If d > Lo
then the spring is always under tension
spring length = ( d*2+x^2)^0.5
spring tension = [( d*2+x^2)^0.5 - Lo ]k
equilibrium at x = 0 and stable
x component of restoring spring force = [( d*2+x^2)^0.5 - Lo ]k * [- x / ( d*2+x^2)^0.5]
so
m d^2x/dt^2 = [( d*2+x^2)^0.5 - Lo ]k * [- x / ( d*2+x^2)^0.5]
for small motion x<<d
m d^2x/dt^2 = [d - Lo ]k * [- x / d] = - k x * (1-Lo/d)
oh my, looks like a plain old spring with a new K = original spring k [ 1 -Lo/d]
Use your old spring equations with that new K
T = 2 pi sqrt { m/k [ 1 -Lo/d] }

For the second case you have an unstable mess. There are equilibrium points at a positive x and at a negative x.