Question

Please solve this question of mechanics?

I am confused and no idea to solve this question

Answer 1

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.