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 ?

Options

A. It will remain unchanged
B. It will rise
C. It will fall
D. First it will and then fall

Answer

To solve the problem of how throwing steel balls from a boat into a tank of water affects the water level, we can break it down into several steps:

### Step 1: Understand the Initial Situation
Initially, the boat is floating on the surface of the water, carrying steel balls. The boat and the balls together displace a certain volume of water, which corresponds to the weight of the boat plus the balls.

### Step 2: Apply Archimedes' Principle
According to Archimedes' principle, the buoyant force acting on the boat is equal to the weight of the water displaced. The volume of water displaced (V1) can be calculated using the formula:
\[ V_1 = \frac{M + n \cdot m}{\rho_w} \]
where \( M \) is the mass of the boat, \( n \) is the number of balls, \( m \) is the mass of one ball, and \( \rho_w \) is the density of water.

### Step 3: Analyze the Effect of Throwing a Steel Ball
When one steel ball is thrown into the water, the total mass of the system (boat + balls) decreases by the mass of the ball. The boat now displaces less water because it has lost the weight of one ball.

### Step 4: Calculate the New Volume Displaced
After throwing one ball, the new volume of water displaced (V2) by the boat can be calculated as:
\[ V_2 = \frac{M + (n-1) \cdot m}{\rho_w} \]
This shows that the volume displaced by the boat decreases because the total mass has decreased.

### Step 5: Compare the Volumes
Since the density of steel is greater than the density of water, the volume of water displaced by the steel ball when it is submerged (V_ball) is less than the volume of water displaced by the weight of the ball when it was in the boat. The volume displaced by the ball when submerged can be calculated as:
\[ V_{ball} = \frac{m}{\rho_{steel}} \]
where \( \rho_{steel} \) is the density of steel.

### Step 6: Conclusion
Since the volume of water displaced by the boat decreases more than the volume displaced by the steel ball when it is submerged, the overall water level in the tank will decrease.

### Final Statement
Therefore, when steel balls are thrown from the boat into the tank one by one, the water level in the tank will fall.

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