Question

A 2.0 kg freshwater fish at the surface of a lake is neutrally buoyant. If the density of its body with its swim bladder deflated is 1060 kg/m3, what volume of gas must be in the swim bladder for the fish to be neutrally buoyant?

Answer 1

The volume of gas required in the swim bladder of the 2.0 kg freshwater fish to achieve neutral buoyancy is approximately 113 mL. This calculation is based on the need for the overall density of the fish to equal that of the surrounding freshwater when the swim bladder is inflated.

Step-by-step explanation:

The question asks what volume of gas must be in the swim bladder of a 2.0 kg freshwater fish for it to be neutrally buoyant, given that the density of its body with the swim bladder deflated is 1060 kg/m3. To answer this, we can use Archimedes' principle, which states that the buoyant force on an object submerged in a fluid is equal to the weight of the fluid that is displaced by the object.

Since the fish is neutrally buoyant, the buoyant force is equal to the gravitational force (weight) of the fish. The amount of water displaced by the fish (and its swim bladder) must have a mass equal to that of the fish itself, which is 2.0 kg.

With the density of freshwater being approximately 1000 kg/m3, the volume of water displaced is the mass divided by the density, which is V = m/ρ = 2.0 kg / 1000 kg/m3 = 0.002 m3 (2 liters).

When the swim bladder is deflated, the volume of water that would have to be displaced for the fish to be neutrally buoyant is the fish's volume minus the volume of the swim bladder. The volume of the fish with deflated swim bladder can be calculated by dividing the mass of the fish by the density of its body, which is Vfish = m/density = 2.0 kg / 1060 kg/m3 ≈ 0.001887 m3.

To achieve neutral buoyancy, the volume of the swim bladder (Vbladder) should be the difference between the total displaced volume and the deflated body volume, which is Vbladder = V - Vfish = 0.002 m3 - 0.001887 m3 ≈ 0.000113 m3 or 113 mL.

Answer 2

To determine the volume of gas needed in the swim bladder for the fish to be neutrally buoyant, we can use Archimedes' principle and the formula for buoyant force. By equating the weight of the fish to the buoyant force, we can calculate the volume of gas in the swim bladder. The volume is found to be 1.9 liters.

Step-by-step explanation:

To determine the volume of gas needed in the swim bladder for the fish to be neutrally buoyant, we need to consider Archimedes' principle and buoyancy. When an object is neutrally buoyant, the buoyant force acting on it equals the weight of the object. The buoyant force is given by the formula Fb = ρVg, where ρ is the density of the liquid (in this case, water), V is the volume of the displaced liquid, and g is the acceleration due to gravity.

For the fish to be neutrally buoyant, the weight of the fish must be equal to the buoyant force. The weight of the fish is given by the formula W = mg, where m is the mass of the fish and g is the acceleration due to gravity. Since the fish is neutrally buoyant, the weight of the fish is equal to the buoyant force, so we can equate the two formulas:

mg = ρVg

Cancelling out g from both sides of the equation, we get:

m = ρV

Since the density of the fish is different from the density of the water, we can rewrite the equation in terms of densities:

m = ρfishVfish = ρwaterV

Dividing both sides of the equation by the density of the fish, we get:

Vfish = V = m/ρfish

Substituting the given values into the equation, we have:

V = 2.0 kg / 1060 kg/m^3 = 0.0019 m^3 = 1.9 L

Therefore, the volume of gas that must be in the swim bladder for the fish to be neutrally buoyant is 1.9 liters.