Fish achieve neutral buoyancy (so they don't have to swim constantly to stay in place) via a swim bladder. A swim bladder is a little internal sack that they can inflate/deflate with air, which changes their volume but not their total mass. To see how this allows them to change their buoyancy, let's consider the situation of a fish floating at rest in the ocean at some arbitrary depth. It inflates its swim bladder and increases its volume to 1.1 times its original volume without losing any mass. It therefore should begin to accelerate upwards automatically without having to swim. What is its acceleration in m/s^2?

Question

Fish achieve neutral buoyancy (so they don't have to swim constantly to stay in place) via a swim bladder. A swim bladder is a little internal sack that they can inflate/deflate with air, which changes their volume but not their total mass. To see how this allows them to change their buoyancy, let's consider the situation of a fish floating at rest in the ocean at some arbitrary depth. It inflates its swim bladder and increases its volume to 1.1 times its original volume without losing any mass. It therefore should begin to accelerate upwards automatically without having to swim. What is its acceleration in m/s^2?

Answer 1

To find the acceleration of the fish, we can use the principle of buoyancy and Newton's second law of motion.

Let's assume that the fish has a mass of m and initially experiences a net force of zero, since it is floating at rest. When the fish inflates its swim bladder and increases its volume to 1.1 times its original volume, the buoyant force acting on the fish will change.

The buoyant force (F_b) on an object immersed in a fluid is equal to the weight of the fluid displaced by the object. In this case, the buoyant force can be calculated as the difference between the weight of the fish and the weight of the fish with the increased volume.

The buoyant force (F_b) can be expressed as:

F_b = ρ_water * g * V_displaced

where ρ_water is the density of water, g is the acceleration due to gravity, and V_displaced is the volume of water displaced by the fish.

Initially, the volume of the fish is V and the buoyant force is equal to the weight of the fish:

F_b_initial = m * g

When the fish inflates its swim bladder and increases its volume to 1.1 times its original volume, the new buoyant force becomes:

F_b_final = ρ_water * g * V_displaced_final

Since the fish does not lose any mass, its weight remains the same:

m * g = ρ_water * g * V_displaced_final

Simplifying the equation, we find:

V_displaced_final = (m * g) / (ρ_water * g)

V_displaced_final = m / ρ_water

Therefore, the buoyant force on the fish after inflation is:

F_b_final = ρ_water * g * (m / ρ_water)

F_b_final = m * g

Since the buoyant force is equal to the weight of the fish, the net force on the fish will be:

F_net = F_b_final - F_b_initial

F_net = m * g - m * g

F_net = 0

Since the net force on the fish is zero, the fish will not accelerate and will remain at rest.