Ring and boat in champagne glass
A romantic sailor proposed to his fiance by placing a ring in a small boat floating in her Champagne glass at dinner. A clumsy waiter bumps the glass and the ring falls to the bottom of the glass while the boat remains floating. How does the level of Champagne in the glass change once the ring sinks?
Answer
This problem is about buoyancy. The buoyant force is equal to the weight of the displaced fluid. In other words, put an object in a fluid and it will displace fluid, causing the level to rise. If an object floats, that means the buoyant force is enough to cancel out the force of gravity on that object. If it sinks, the object isn't able to displace enough fluid to cancel out gravity.
For this problem, it will be helpful to draw free body diagrams in the before and after case. Before, you have the boat+ring combination floating and two forces acting on it: gravity and buoyancy. Remember that the force of gravity is for the combined masses, boat and ring. After, you have the boat still floating, but the ring on the bottom of the glass. Like we noted earlier, the ring sinks because the buoyant force is not enough to cancel out gravity. In this case the normal force from the glass does the rest of the job.
Now let's start writing some equations. In the Before case, we have the force of gravity equal to the buoyant force:
ρ * V_both * g = (m_boat + m_ring) * g
where V_both is the volume of fluid displaced by the boat with the ring in it.
In the After case, we have a similar equation for the boat:
ρ * V_boat * g = m_boat * g
and we can write an inequality for the ring because we know the buoyant force is weaker than gravity:
ρ * V_ring * g < m_ring * g
Now let's try to solve for something useful. Adding the last two expressions gives another inequality:
ρ * (V_boat + V_ring) * g < (m_boat + m_ring) * g
and now we can plug in the first equation:
ρ * (V_boat + V_ring) * g < ρ * V_both * g
and arrive at the conclusion:
V_boat + V_ring < V_both
in other words, the volume of fluid displaced by the boat with the ring in it is greater than the total volume displaced when the objects are separate. So when the ring sinks, less fluid is being displaced so the water level lowers.
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This problem is about buoyancy. The buoyant force is equal to the weight of the displaced fluid. In other words, put an object in a fluid and it will displace fluid, causing the level to rise. If an object floats, that means the buoyant force is enough to cancel out the force of gravity on that object. If it sinks, the object isn't able to displace enough fluid to cancel out gravity.
For this problem, it will be helpful to draw free body diagrams in the before and after case. Before, you have the boat+ring combination floating and two forces acting on it: gravity and buoyancy. Remember that the force of gravity is for the combined masses, boat and ring. After, you have the boat still floating, but the ring on the bottom of the glass. Like we noted earlier, the ring sinks because the buoyant force is not enough to cancel out gravity. In this case the normal force from the glass does the rest of the job.
Now let's start writing some equations. In the Before case, we have the force of gravity equal to the buoyant force:
ρ * V_both * g = (m_boat + m_ring) * g
where V_both is the volume of fluid displaced by the boat with the ring in it.
In the After case, we have a similar equation for the boat:
ρ * V_boat * g = m_boat * g
and we can write an inequality for the ring because we know the buoyant force is weaker than gravity:
ρ * V_ring * g < m_ring * g
Now let's try to solve for something useful. Adding the last two expressions gives another inequality:
ρ * (V_boat + V_ring) * g < (m_boat + m_ring) * g
and now we can plug in the first equation:
ρ * (V_boat + V_ring) * g < ρ * V_both * g
and arrive at the conclusion:
V_boat + V_ring < V_both
in other words, the volume of fluid displaced by the boat with the ring in it is greater than the total volume displaced when the objects are separate. So when the ring sinks, less fluid is being displaced so the water level lowers.