Please solve this question of mechanics?

enter image source here I am confused and no idea to solve this question

1 Answer
Feb 22, 2018

The stable point is x_2

Explanation:

Considering the movement axis origin at x_1, a mass m particle movement subjected to a force f(x) is described by

m ddot x = f(x)

where f(x) is an odd function like k x

Solving for x we have

m ddot x - k x = 0

multiplying by dot x we have

m dot x ddot x -k x dot x = 0 or

1/2m d/(dt)(dot x)^2-k 1/2d/(dt) x^2 = 0

then in the case of (x_1) we have

1/2m(dot x)^2 = C_0+1/2k x^2 or

m(dot x)^2-k x^2 = 2C_0 which characterizes unbounded orbits (hyperbolic)

This means also that the kinetic energy increases as the point goes away from x_1 denoting instability.

Analogously at point (x_2) we have

f(x) = -kx rArr 1/2m d/(dt)(dot x)^2+k 1/2d/(dt) x^2 = 0

and then

1/2m(dot x)^2 = C_0-1/2k x^2 or

m(dot x)^2 +k x^2=2C_0

which defines a center or harmonic movement around (x_2)

With dissipation the point finishes the movement at x_2 characterizing mechanical stability.