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A chest full of gold goes down with a pirate ship. As it sinks, the pirate ship breaks up, and the chest slips out of the wreckage. The chest is a cube 50 cm on each side. The chest is oriented upright in the water, so that its top surface is 20 m below the waves.
Question 1: What is the pressure of the water on the top of the chest? What is the pressure of the water on the bottom of the chest?
Answer: The top surface is at a depth of d = 20m below the surface. Pressure = (density)*(g)*(depth) = (1000 kg/m^3)*(9.8 m/s^2)*(20 m) = 196,000 N/m^2 = 196,000 pa The bottom surface is at a depth of d = 20.5 m below the surface Pressure = (density)*(g)*(depth) = (1000 kg/m^3)*(9.8 m/s^2)*(20.5 m) = 200,900 N/m^2 = 200,900 pa
Question 2: What is the force exerted by water on the top of the chest? What is the force exerted by water on the bottom of the chest?
Answer: Force on a surface = pressure * area. The area of each side of the chest is Area = (length)*(width) = (0.5 m)*(0.5 m) = 0.25 m^2 Top surface: force = (196,000 N/m^2)*(0.25 m^2) = 49,000 N downwards Bottom surface: force = (200,900 N/m^2)*(0.25 m^2) = 50,225 N upwards
Question 3: The chest has a mass of 2200 kg. What is the net force on it in the vertical direction? Does it sink?
Answer: The net vertical force on the chest is made of three pieces: let's call downward forces negative, upward forces positive. Net force = Force of gravity (down) + force of water on top (down) + force of water on bottom (up) = - (mass)*(g) - 49,000 N + 50,225 N = - (2000 kg)*(9.8 m/s^2) - 49,000 N + 50,225 N = - 21,560 N - 49,000 N + 50,225 N = -20,335 N The net force is downwards, therefore the chest sinks. Another way to solve this problem is to realize that the upwards buoyant force is equal to the mass of water which the chest displaces. The volume of the chest is volume = (0.5 m)*(0.5 m)*(0.5 m) = 0.125 m^3 Net force = (buoyant force up) - (gravitational force down) = (mass of water displaced)*(g) - (mass of chest)*(g) = (1000 kg/m^3)*(0.125 m^3)*(9.8 m/s^2) - (2200 kg)*(9.8m/s^2) = 1225 N up - 21,560 N down = -20,335 N (down) Which is exactly the same as the force calculated above. It is _also_ true that the density of the chest is density = (mass)/(volume) = (2000 kg)/(0.125 m^3) = 16,000 kg/m^3 which is larger than water's density of 1,000 kg/m^3. Therefore, the chest must sink.
This page maintained by Michael Richmond. Last modified Dec 16, 1997.
Copyright © Michael Richmond. This work is licensed under a Creative Commons License.