2018 A/L Physics Past Paper · Question 29 (English Medium)

Answer: (2) Initially, $C$ starts to contribute to the flow at the outlet followed by $B$ and then $A$ in succession.

What the question asks

The tanks empty slowly.

We want: in what order do they start to send water to the outlet?

Relevant theory

Consider a point A at a depth h below the surface of a liquid exposed to the atmosphere.

In still water, the pressure grows with depth:

$$p = p_0 + h\rho g$$

So the pressure in the pipe is set by how high the open water stands above it.

Required equations

$p_{top} = p_0 + \rho g\,(h_{tank} - y_{valve})$

$p_{bottom} = p_0 + \rho g\,(h_{C} - y_{valve})$

$p_{top} > p_{bottom} \iff h_{tank} > h_{C}$

Reading the options

Each option gives a different order.

Follow the stages one by one. Then pick the option.

Reasoning and calculation

Stage 1: the start

$C$

$C$ has the highest water. Its pipe has no valve.

So $C$ sets the pressure in the common pipe.

At $V_1$ and $V_2$ the push from below is bigger than the push from above. So both valves stay shut.

Only $C$ sends water.

$C$ empties, so its level falls.

When it falls to the level of $B$, the pushes on both sides of $V_2$ are equal.

A little lower, and $V_2$ opens.

Stage 2: $C$ reaches the level of $B$

$C + B$

Now $B$ and $C$ send water together. They share one level, and it falls slowly.

$A$ is still lower. So $V_1$ stays shut.

When the shared level falls to the level of $A$, $V_1$ opens too.

Now all three send water together.

Stage 3: the level reaches $A$

$C + B + A$

Both valves are open.

All three tanks send water at the same level.

The order was: $C$, then $C$ and $B$, then $C$, $B$ and $A$.

The order we found: first $C$, then $B$, last $A$.

Only option (2) says this order.

Checks

Check.

At every moment, the highest open tank sets the pressure in the pipe.

All open tanks stand at the same level.

The valves never let water go up into $A$ or $B$. So their levels never rise.

Conclusion

First only $C$ sends water.

When its level reaches $B$, $B$ joins.

When the level reaches $A$, $A$ joins.

The answer is option (2).

Thinking that all the tanks flow together from the start.

A one-way valve opens only when the push above it is bigger than the push below.

The highest open water sets the pressure in the pipe.

Explore

At the start the pipe is filled from $C$, the highest level, and $V_1$ and $V_2$ stay shut.

Only $C$ feeds the outlet, and its level slowly falls.

When it reaches the level of $B$, $V_2$ opens and $B$ starts to feed too.

When the common level reaches the level of $A$, $V_1$ opens as well: first $C$, then $B$, then $A$.