Final answer:
A 6.0 kg freshwater fish needs 0.34 liters of gas in its swim bladder to be neutrally buoyant. This is calculated by considering the fish's density without gas in the bladder, the density of freshwater, and using Archimedes' Principle to equalize the fish's average density with that of the surrounding water. Therefore, the volume of gas needed in the swim bladder for the fish to be neutrally buoyant is 0.34 liters.
Step-by-step explanation:
When a 6.0 kg freshwater fish is neutrally buoyant, it means the buoyant force acting on it by the water equals its weight. The buoyant force can be determined through Archimedes' Principle, which states that the buoyant force is equal to the weight of the fluid displaced by the object. Since we are given the fish's deflated swim bladder density which is 1060 kg/m³, the first step is to calculate the volume of the fish body without the swim bladder inflated.
To find the volume of the fish, we use the formula for density, ρ = m/V, where ρ is density, m is mass, and V is volume. Rearranging the formula to solve for volume gives us V = m/ρ. Substituting the given values yields V = 6.0 kg / 1060 kg/m³. Upon calculation, V = 0.00566 m³ or 5.66 L.
Since the fish is neutrally buoyant, we know the volume of water displaced by the fish equals the volume of the fish itself. The density of freshwater is approximately 1000 kg/m³, so the volume of the swim bladder gas must be such that the average density of the fish with the inflated swim bladder equals the density of freshwater.
If we let Vb be the volume of the bladder, the total volume of the fish Vt is V + Vb. The total mass stays the same, so the fish's new average density with the gas-filled bladder would be ρ = m/Vt. To be neutrally buoyant, ρ must equal 1000 kg/m³ (freshwater's density). Setting this up and solving for Vb gives us 6.0 kg / (5.66 L + Vb L) = 1000 kg/m³. The resulting equation simplifies to Vb L = 6.0 L - 5.66 L, hence Vb = 0.34 L.
Therefore, the volume of gas needed in the swim bladder for the fish to be neutrally buoyant is 0.34 liters.