A boat carrying steel balls is floating on the surface of water in a tank. If the balls are thrown into the tank one by one, how will it affect the level of water

The correct answer is: it will fall

To understand why the water level falls, we can analyze the volume of water displaced in two scenarios: when the steel balls are inside the floating boat, and when the steel balls are thrown into the tank and sink.

Case 1: When the boat carrying the steel balls is floating
According to Archimedes' principle, any floating object displaces a weight of fluid equal to its own weight. Let:
- $M_b$ be the mass of the boat,
- $M_s$ be the total mass of the steel balls,
- $d_w$ be the density of water, and
- $d_s$ be the density of steel.

Since the boat and the steel balls are floating together as a single system, the total weight of the system is supported by buoyancy. The volume of water displaced ($V_1$) is given by:

$V_1=\frac{M_b+M_s}{d_w}$

Case 2: When the steel balls are thrown into the water
When the steel balls are dropped into the tank, they sink to the bottom because the density of steel is greater than the density of water ($d_s>d_w$). The boat continues to float on its own.
The total volume of water displaced in this case ($V_2$) is the sum of:
1. The water displaced by the floating boat: $\frac{M_b}{d_w}$
2. The water displaced by the submerged steel balls, which is equal to their own physical volume: $\frac{M_s}{d_s}$

So, the new total volume of water displaced is:

$V_2=\frac{M_b}{d_w}+\frac{M_s}{d_s}$

Comparing the two volumes of displaced water:
Let us find the difference between the initial displaced volume and the final displaced volume:

$V_1-V_2=\left(\frac{M_b}{d_w}+\frac{M_s}{d_w}\right)-\left(\frac{M_b}{d_w}+\frac{M_s}{d_s}\right)$

$V_1-V_2=M_s\left(\frac{1}{d_w}-\frac{1}{d_s}\right)$

Since steel is denser than water ($d_s>d_w$), it follows that:

$\frac{1}{d_w}>\frac{1}{d_s}$

Therefore, the difference $V_1-V_2$ is positive, which means:

$V_1>V_2$

Because the total volume of water displaced decreases when the steel balls sink, the water level in the tank will fall.