Can Conservation of Energy Explain Pulley and Inclined Plane Mechanics?

[showzen]

Homework Statement

Homework Equations

Conservation of Energy / Virtual Work
$$\sum_i m_i g h_i = 0$$

The Attempt at a Solution

Moving the pulley I labeled A down by $\delta y$ moves $w$ down by $\delta y$, and moves $W$ up by $\frac{1}{2} sin\theta$.

So by conservation of energy I have $$-w\delta y + \frac{1}{2}W \delta y sin\theta = 0$$ leading to $W = \frac{2w}{sin\theta}$.

I feel uncertain about this answer however, because it seems like the mechanics would be the same if you removed Pulley A and replaced it with $w$.

 

[haruspex]

Moving the pulley I labeled A down by δy moves w down by δy

Are you sure about that?

 

[kq6up]

Not sure why you are using virtual work either. I would attack this problem with mechanical advantage of a system of pulleys.

Regards,
KQ6UP

 

[showzen]

Are you sure about that?

My reasoning here is that if pulley A is moved down to be even with the anchor point, $w$ will have dropped by the initial length between the anchor and the pully.

 

[showzen]

My reasoning here is that if pulley A is moved down to be even with the anchor point, $w$ will have dropped by the initial length between the anchor and the pully.View attachment 104192

I see my mistake now! $\delta y$ moves $w$ by $2\delta y$. So $W=\frac{4w}{sin\theta}$, which is what I get from force analysis as well.

 

[haruspex]

I see my mistake now! $\delta y$ moves $w$ by $2\delta y$. So $W=\frac{4w}{sin\theta}$, which is what I get from force analysis as well.

That looks good.