A lost shipping container is found resting on the ocean floor and completely submerged. The container is 5.7 m long, 2.7 m wide, and 2.3 m high. Salvage experts attach a spherical balloon to the top of the container and inflate it with air pumped down from the surface. When the balloon's radius is 1.3 m, the shipping container just begins to rise towards the surface. What is the mass of the container? Ignore the mass of the balloon and the air within it. Do not neglect the buoyant force exerted on the shipping container by the water. The density of seawater is 1025 kg/m3.
To find the mass of the container, we can use Archimedes' principle, which states that the buoyant force on an object submerged or floating in a fluid is equal to the weight of the fluid displaced by the object.
The buoyant force can be calculated as follows:
Buoyant force = weight of the displaced fluid
= density of fluid * volume of displaced fluid * acceleration due to gravity
In this case, the fluid is seawater and the displaced fluid is the volume of the container. The volume of the container can be calculated using its dimensions:
Volume of container = length * width * height
Let's calculate the volume of the container first:
Volume of container = 5.7 m * 2.7 m * 2.3 m
= 36.747 m^3
Now, we can calculate the buoyant force:
Buoyant force = 1025 kg/m^3 * 36.747 m^3 * 9.8 m/s^2
≈ 363,008.695 N
Since the balloon just begins to lift the container, the buoyant force must be equal to the weight of the container and its contents.
The weight of an object can be calculated using the formula:
Weight = mass * acceleration due to gravity
Therefore, we can find the mass of the container:
Mass = Weight / acceleration due to gravity
= Buoyant force / acceleration due to gravity
= 363,008.695 N / 9.8 m/s^2
≈ 37,003.95 kg
So, the mass of the container is approximately 37,003.95 kg.
To determine the mass of the container, we need to consider the buoyant force exerted on it by the water. The buoyant force is equal to the weight of the water displaced by the container.
First, let's find the volume of the shipping container. The volume can be calculated by multiplying its length, width, and height:
Volume = Length x Width x Height
Volume = 5.7 m x 2.7 m x 2.3 m
Volume = 34.515 m³
Now let's find the weight of the water displaced by the container. We can calculate this by multiplying the volume of water displaced by the density of seawater (1025 kg/m³). The weight of the water displaced is then equal to the buoyant force acting on the container.
Weight of water displaced = Volume x Density of seawater
Weight of water displaced = 34.515 m³ x 1025 kg/m³
Weight of water displaced ≈ 35363.875 kg
Next, let's calculate the net buoyant force acting on the container. The net buoyant force is equal to the weight of the water displaced minus the weight of the container.
Net Buoyant Force = Weight of water displaced - Weight of container
Considering that the container is just beginning to rise, its weight is equal to the net buoyant force:
Weight of container = Net Buoyant Force
Weight of container = 35363.875 kg
Therefore, the mass of the container is approximately 35363.875 kg.