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Chapter 3: Problem 111

Consider a conical funnel held upside down and submerged slowly in a container of water. Discuss the force needed to submerge the funnel if the spout is open to the atmosphere. Compare with the force needed to submerge the funnel when the spout opening is blocked by a rubber stopper.

Short Answer

Expert verified
The force required to submerge the funnel with an open spout is less than the force required when the spout is blocked, due to the additional pressure from the trapped air that needs to be overcome in the latter case.

Step by step solution

01

Analyze the situation when the spout is open

In the case of the spout being open, the water can flow freely into the funnel. So, the force required would be just the force required to push the conical part into the water, which is simply equal to the weight of the water displaced by the funnel. According to Archimedes' principle, the buoyant force of the water displaced by the conical part of the funnel is equal to the weight of that water, say \(F1\).
02

Analyze the situation when the spout is blocked

When the spout is blocked, air inside the funnel is trapped. The volume of the air inside the funnel resists being compressed, thereby creating additional pressure. Now, to submerge the funnel, that additional pressure (P) must be overcome. This pressure will equate to the pressure of the water at the depth of the blocked spout. This force will be given by \(F2 = P*A\), where A is the area of the spout.
03

Compare the forces

The force needed to submerge the funnel when the spout is blocked is certainly more than when the spout is open, because in the second case it is necessary to overcome not only the buoyant force \(F1\), but also the additional force due to the trapped air \(F2\). Therefore, the total force in case two would be \(F = F1 + F2\).

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