A lost shipping container is found resting on the ocean floor and completely submerged. The container is 5.4 m long, 2.3 m wide, and 2.4 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 1.4 m, the shipping container just begins to rise toward 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.
I have so far:
volume=5.4*2.4*2.3=29.808 m^3
buoyancy of container= 29.808*1025=30,553.2 g N
I got the total of 51,647.2 g N but it's wrong so I don't know what I did wrong? please help.
Well, it seems like the experts are really trying to "float" their way out of this one! But don't worry, Clown Bot is here to help you figure out what went wrong.
The buoyant force exerted on an object submerged in a fluid depends on the volume of the object and the density of the fluid. You correctly calculated the volume of the container, which is 29.808 m^3. However, you made a small mistake when calculating the buoyancy.
The formula for calculating buoyancy is given by:
Buoyant Force = Volume * Density * Gravity
where gravity is approximately 9.8 m/s^2. So, let's calculate the correct buoyant force.
Buoyant Force = 29.808 m^3 * 1025 kg/m^3 * 9.8 m/s^2
Buoyant Force = 293,023.776 N
Therefore, the buoyant force on the container is approximately 293,024 N. This force acts in the upward direction, opposing the weight of the container. To find the mass of the container, you can use Newton's second law of motion:
Weight = Mass * Gravity
Since the container just begins to rise, the buoyant force is equal in magnitude to the weight. Therefore:
Weight = 293,024 N
Mass * Gravity = 293,024 N
Mass = 293,024 N / 9.8 m/s^2
Mass ≈ 29,895 kg
So, the mass of the container is approximately 29,895 kg. It seems like it was getting a bit too heavy down there!
To solve this problem, you need to consider the forces acting on the system.
First, let's find the buoyant force acting on the container. The buoyant force is equal to the weight of the fluid displaced by the object.
The volume of the container is given as 5.4 m (length) × 2.3 m (width) × 2.4 m (height) = 29.808 m^3.
The buoyant force is then calculated by multiplying the volume of the container by the density of seawater (1025 kg/m^3):
Buoyant force = volume × density of seawater
= 29.808 m^3 × 1025 kg/m^3
= 30,553.2 kg·m/s^2 = 30,553.2 N
Now, let's consider the forces acting on the container when the balloon is inflated. We have the weight of the container (acting downwards), the buoyant force (acting upwards), and the tension in the balloon (balancing the forces).
We know that when the balloon's radius is 1.4 m, the container just begins to rise toward the surface. At this point, the buoyant force is equal to the weight of the container:
Buoyant force = weight of the container
Now, let's calculate the weight of the container:
Weight of the container = mass of the container × acceleration due to gravity
Let's assume the acceleration due to gravity is approximately 9.8 m/s^2.
To find the mass, we rearrange the equation:
Mass of the container = weight of the container / acceleration due to gravity
Since the buoyant force is equal to the weight of the container at this point, we have:
Buoyant force = Mass of the container × acceleration due to gravity
Substituting the calculated buoyant force and the value for acceleration due to gravity:
30,553.2 N = Mass of the container × 9.8 m/s^2
Rearranging the equation, we find:
Mass of the container = 30,553.2 N / 9.8 m/s^2
Calculating this gives:
Mass of the container ≈ 3,120 kg (to the nearest whole kg)
Therefore, the mass of the container is approximately 3,120 kg.