Solution
Step 1: Analyzing the Initial Equilibrium
Let the area of piston $P$ be $A$. Since the mass of $Q$ is $\alpha m$ and they stay in equilibrium at the same level initially, the pressures exerted by the pistons must balance. Since pressure $P = \frac{\text{Force}}{\text{Area}}$:
$$ \frac{mg}{A} = \frac{\alpha mg}{A_Q} \implies A_Q = \alpha A $$
Thus, the area of vessel $Q$ is $\alpha$ times the area of vessel $P$.
Step 2: Force $F$ applied on Piston P
When force $F$ is applied to $P$, piston $Q$ moves up by height $h$. Due to conservation of volume of the incompressible liquid:
$$ \text{Volume pushed down} = \text{Volume pushed up} $$
$$ A \cdot x_p = (\alpha A) \cdot h \implies x_p = \alpha h $$
The difference in liquid levels becomes $H_{total} = x_p + h = \alpha h + h = h(1+\alpha)$.
Balancing pressure at the level of piston $P$:
$$ \frac{F + mg}{A} = \frac{\alpha mg}{\alpha A} + \rho g h(1+\alpha) $$
Canceling the weight terms ($mg/A$ on both sides):
$$ \frac{F}{A} = \rho g h (1+\alpha) \quad \dots (1) $$
Step 3: Force $\beta F$ applied on Piston Q
Now, force is removed from $P$ and $\beta F$ is applied to $Q$. Piston $P$ will rise by some height $H’$, and $Q$ will move down by $x_q$.
By volume conservation:
$$ A \cdot H’ = (\alpha A) \cdot x_q \implies x_q = \frac{H’}{\alpha} $$
The total level difference is $H’ + x_q = H’ (1 + \frac{1}{\alpha}) = H’ (\frac{\alpha + 1}{\alpha})$.
Balancing pressure at the level of piston $Q$ (since $P$ is higher):
$$ \frac{\beta F + \alpha mg}{\alpha A} = \frac{mg}{A} + \rho g H’ \left( \frac{\alpha + 1}{\alpha} \right) $$
Canceling weights again:
$$ \frac{\beta F}{\alpha A} = \rho g H’ \left( \frac{\alpha + 1}{\alpha} \right) \quad \dots (2) $$
Step 4: Solving for the final height $H’$
Substitute the value of $F/A$ from equation (1) into equation (2):
$$ \frac{\beta}{\alpha} \left[ \rho g h (1+\alpha) \right] = \rho g H’ \left( \frac{\alpha + 1}{\alpha} \right) $$
Canceling common terms $\rho$, $g$, $(1+\alpha)$, and $\alpha$:
$$ \beta h = H’ $$
Answer: The piston P moves up by a height $\beta h$.