Conservation of energy problem: Two masses, a pulley and an incline

[Tassandro]

Homework Statement

Two material points of masses M and m are joined together by means of a wire that passes in A through a fixed pulley. The mass m hangs vertically; the largest M rests on a smooth inclined plane that forms an $\alpha$ angle with the vertical. M starts its movement, sliding along the plane, without initial speed, starting from point B0 located on the vertical of A. Demonstrate that point M performs amplitude oscillations $x = \overline {B_oB} = \frac {2m (M-m) h \cos \alpha} {m ^ 2-M ^ 2\cos ^ 2 \alpha}$ where $h = \overline {B_oA}$ and fulfilling the condition $M \cos \alpha < m < M$

Relevant Equations

Gravitational energy: $mgh$

If M moves $x$ along the plane, her height variation in $x \cos(\alpha)$, and, but I don't know how to find the variation of the height of $m$

 

[SammyS]

Homework Statement:: Two material points of masses M and m are joined together by means of a wire that passes in A through a fixed pulley. The mass m hangs vertically; the largest M rests on a smooth inclined plane that forms an alpha angle with the vertical. M starts its movement, sliding along the plane, without initial speed, starting from point B0 located on the vertical of A. Demonstrate that point M performs amplitude oscillations $x = \overline {B_oB} = \frac {2m (M-m) h \cos \alpha} {m ^ 2-M ^ 2\cos ^ 2 \alpha}$ where $h = \overline {B_oA}$ and fulfilling the condition $M \cos \alpha < m < M$
Relevant Equations:: Gravitational energy: $mgh$

View attachment 265869View attachment 265869
If M moves \(\displaystyle x\) along the plane, her height varation in \(\displaystyle x\cosα[\MATH], and, but I dont't know how to find the variation of the height of \(\displaystyle m\) :/\)

Hello @Tassandro .


Consider the change in the length of wire from the mass, M to the pulley, when x = 0, compared to its length for an arbitrary value of x.

 

[berkeman]

Homework Statement:: Two material points of masses M and m are joined together by means of a wire that passes in A through a fixed pulley. The mass m hangs vertically; the largest M rests on a smooth inclined plane that forms an alpha angle with the vertical. M starts its movement, sliding along the plane, without initial speed, starting from point B0 located on the vertical of A. Demonstrate that point M performs amplitude oscillations $x = \overline {B_oB} = \frac {2m (M-m) h \cos \alpha} {m ^ 2-M ^ 2\cos ^ 2 \alpha}$ where $h = \overline {B_oA}$ and fulfilling the condition $M \cos \alpha < m < M$
Relevant Equations:: Gravitational energy: $mgh$

View attachment 265869View attachment 265869
If M moves \(\displaystyle x\) along the plane, her height varation in \(\displaystyle x\cosα[\MATH], and, but I dont't know how to find the variation of the height of \(\displaystyle m\) :/\)

Is that bottom plane free to rotate about point $B_0$ like a see-saw? Is that why you can get an oscillatory motion?

 

[haruspex]

Is that bottom plane free to rotate about point $B_0$ like a see-saw? Is that why you can get an oscillatory motion?

No, it's because there is an equilibrium position

Spoiler

given by $Mg\cos(\alpha)=mg\cos(\alpha-\beta)$

, where beta is the angle of the string to the vertical.