Leverage and Forces on a Pivot System

[Saints-94]

If the weight of the bin could be determined, could an equation be formed to work out the longest the lever could be without the bin tipping?

 

[sophiecentaur]

So the equation above is not relevant as the angle does not need to be considered as the force is worked out using a ratio of X and Y?

For a vertically applied force, you are dealing with similar triangles here. The ratio of lengths of the horizontal sides are the same as the ratio of measurements along the bar.
X and X+Y, actually

If the weight of the bin could be determined, could an equation be formed to work out the longest the lever could be without the bin tipping?

Yes. consider the edge of the bin as a fulcrum and the moments of the applied force and the total weight of bin plus rubbish plus lever system. Then equate the moments to give you the longest lever for equilibrium.
However, the user could probably exceed this value by adjusting the angle of the push and 'steadying' the system to avoid tipping.