A thick-walled boat with vertical walls and a hole in the bottom floats freely in the lake for quite a long time. Then the hole is plugged from the outside and a log is lowered inside the boat. Will the water level in the boat rise or fall after this relative to the water level in the lake? Why?
Decision:
Let $m$ be the mass of the log, $S_{1}$ be the area of the bottom of the boat outside and $S_{2}$ be the area of the bottom of the boat inside. According to the condition of the problem
$S_{1}>S_{2}$. (1)
When a log is placed in the boat, it, due to the increase in the total weight of the system, sinks deeper (by some value $h_{1}$) into the lake, thus displacing an additional volume of water
$V = h_{1}S_{1}$, (2)
the mass of which is equal to the mass of the log $m$. The log, sinking into the water filling the boat, displaces the same volume of liquid $V$. As a result, the water level in the boat rises relative to its bottom by some value $h_{2}$ - such that
$V=h_{2}S_{2}$. (3)
From (2) and (3) we obtain:
$\frac{h_{1}}{h_{2}}=\frac{S_{2}}{S_{1}}$,
whence, by virtue of inequality (1), it follows that $h_{2}>h_{1}$. Thus, after the log is submerged, the water level in the boat will be higher than the water level in the lake by an amount equal to the difference $h_{2}-h_{1}$.
Decision:
Let $m$ be the mass of the log, $S_{1}$ be the area of the bottom of the boat outside and $S_{2}$ be the area of the bottom of the boat inside. According to the condition of the problem
$S_{1}>S_{2}$. (1)
When a log is placed in the boat, it, due to the increase in the total weight of the system, sinks deeper (by some value $h_{1}$) into the lake, thus displacing an additional volume of water
$V = h_{1}S_{1}$, (2)
the mass of which is equal to the mass of the log $m$. The log, sinking into the water filling the boat, displaces the same volume of liquid $V$. As a result, the water level in the boat rises relative to its bottom by some value $h_{2}$ - such that
$V=h_{2}S_{2}$. (3)
From (2) and (3) we obtain:
$\frac{h_{1}}{h_{2}}=\frac{S_{2}}{S_{1}}$,
whence, by virtue of inequality (1), it follows that $h_{2}>h_{1}$. Thus, after the log is submerged, the water level in the boat will be higher than the water level in the lake by an amount equal to the difference $h_{2}-h_{1}$.