Question

When in a small pond a person in rowboat, throws an anchor overboard, what happens to the water level?

• A. Goes down
• B. Goes up
• C. First goes up and then goes down
• D. Remains same

Hint:

The key insight is that when the anchor is in the boat, it contributes its full *weight* to water displacement. When it's at the bottom, it only displaces its own *volume*. Since the anchor is denser than water, the volume of water equivalent to its weight is much larger than its actual volume.

Answer 1

The Correct Option is A

Step 1: Analyze the initial state (anchor in the boat). The boat and the anchor together are floating. According to Archimedes' principle, a floating object displaces a volume of water with a weight equal to the total weight of the object. Let \(W_{boat}\) be the weight of the boat and \(W_{anchor}\) be the weight of the anchor. The total weight is \(W_{total} = W_{boat} + W_{anchor}\). The weight of the displaced water is \(W_{disp,1} = W_{total}\). The volume of displaced water is \(V_{disp,1} = \frac{W_{disp,1}}{\rho_{water}g} = \frac{W_{boat} + W_{anchor}}{\rho_{water}g}\).
Step 2: Analyze the final state (anchor in the water). The boat is now floating by itself, and the anchor is fully submerged at the bottom of the pond. - The boat displaces a volume of water corresponding to its own weight: \(V_{boat,disp} = \frac{W_{boat}}{\rho_{water}g}\). - The anchor is submerged and displaces a volume of water equal to its own volume. The volume of the anchor is \(V_{anchor} = \frac{W_{anchor}}{\rho_{anchor}g}\). The total volume of water displaced in the final state is the sum of these two volumes: \[ V_{disp,2} = V_{boat,disp} + V_{anchor} = \frac{W_{boat}}{\rho_{water}g} + \frac{W_{anchor}}{\rho_{anchor}g} \]
Step 3: Compare the initial and final displaced volumes. We compare \(V_{disp,1}\) and \(V_{disp,2}\). \[ V_{disp,1} = \frac{W_{boat}}{\rho_{water}g} + \frac{W_{anchor}}{\rho_{water}g} \] The difference between the two states is the volume displaced by the anchor's weight. We are comparing \(\frac{W_{anchor}}{\rho_{water}g}\) (when it's part of the floating system) with \(\frac{W_{anchor}}{\rho_{anchor}g}\) (when it's submerged). Since an anchor is made of a dense material (e.g., iron), its density \(\rho_{anchor}\) is much greater than the density of water \(\rho_{water}\). Therefore, \(\frac{1}{\rho_{water}}>\frac{1}{\rho_{anchor}}\), which implies \(\frac{W_{anchor}}{\rho_{water}g}>\frac{W_{anchor}}{\rho_{anchor}g}\). This means \(V_{disp,1}>V_{disp,2}\).
Step 4: Conclude the effect on the water level. Since the total volume of water displaced by the system decreases when the anchor is thrown overboard, the overall water level in the pond must go down.