# Energy combined with Circular Motion

## Homework Statement

Consider a small box of mass m sitting on a wedge with an angle θ and fixed to a spring
with a spring constant k and a length in a non-stretched state L. The wedge
rotates with an angular velocity ω around the vertical axis. Find the equilibrium
position of the box and discuss the conditions when such equilibrium is possible and when it is impossible. The box can move only in the direction along the wedge slope and cannot move in the perpendicular direction (e.g. it is on a rail)

## The Attempt at a Solution

I tried by solving total energy at point A (where the spring is unstretched) = total energy at point B (where the spring is streched by a maximum amount) i.e.:

[x is the extension of the spring down the slope]
[va, vb = velocities at point A and B respectively]

1/2m(va)^2 = -mg(L + x)sinθ + 1/2m(vb)^2 + 1/2kx^2

Then I tried to solve using circular motion and F = ma by taking up the slope as my positive direction:

[T denotes tension in the spring]

T - mgsinθ = mL(ω)^2

where ω = v/L

then I just assumed there is no tension when the spring is unstretched (?)

and came up with va = -gsinθ/L

Don't know how to carry on...

I'm seeing this question for the second time in two years and it's super frustrating

Thank you so much!

Ibix