Energy combined with Circular Motion

[phyzz]

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]

Your problem description isn't complete, which makes it a little hard to comment precisely. Is the bottom or the top of the ramp at the axis of rotation? Is the spring attached to the top or bottom of the ramp? Is θ the angle to the horizontal or to the vertical?

That said, I think that energy conservation isn't particularly helpful here - there must be a motor or some such doing work on the system to make it spin, so you can't conserve energy without knowing what that's doing. Better to use the forces, Luke.

Why did you assume that the spring wasn't stretched? Surely it would be either stretched or compressed by the rotation. Are you familiar with the expression for the tension in a spring in terms of its extension (Hooke's Law)? Look it up if necessary, then sub into your force equation. Think about whether or not you need to modify the other terms.

Post your working and the answers to the questions in my first paragraph if you need any more help.