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: Initial state analysis (anchor with boat). The combined boat and anchor float. According to Archimedes' principle, a floating body displaces water equal to its total weight. Let \(W_{boat}\) be the boat's weight and \(W_{anchor}\) be the anchor's weight. The total weight is \(W_{total} = W_{boat} + W_{anchor}\). The weight of 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: Final state analysis (anchor in water). The boat now floats alone, and the anchor rests fully submerged at the bottom. - The boat displaces water equal to its own weight: \(V_{boat,disp} = \frac{W_{boat}}{\rho_{water}g}\). - The submerged anchor displaces water equal to its own volume. The anchor's volume is \(V_{anchor} = \frac{W_{anchor}}{\rho_{anchor}g}\). The total displaced water volume in this state is the sum of these volumes: \[ V_{disp,2} = V_{boat,disp} + V_{anchor} = \frac{W_{boat}}{\rho_{water}g} + \frac{W_{anchor}}{\rho_{anchor}g} \]
Step 3: Comparison of 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 initial and final states lies in the volume displaced by the anchor. We compare \(\frac{W_{anchor}}{\rho_{water}g}\) (when part of the floating system) with \(\frac{W_{anchor}}{\rho_{anchor}g}\) (when submerged). Since anchors are typically made of dense materials (e.g., iron), their density \(\rho_{anchor}\) significantly exceeds the density of water \(\rho_{water}\). Consequently, \(\frac{1}{\rho_{water}}>\frac{1}{\rho_{anchor}}\), leading to \(\frac{W_{anchor}}{\rho_{water}g}>\frac{W_{anchor}}{\rho_{anchor}g}\). This indicates that \(V_{disp,1}>V_{disp,2}\).
Step 4: Conclusion on water level. As the total volume of water displaced by the system diminishes when the anchor is thrown overboard, the overall water level in the pond will decrease.