Fish control their buoyancy with a gas-filled organ called a swim bladder. The a

Question

Fish control their buoyancy with a gas-filled organ called a swim bladder. The average density of a particular fish's tissues, not including gas in the bladder, is 1070 kg/m^3.

If the fish's mass is 10.4 kg, what volume of gas in its swim bladder will keep it in neutral buoyancy-neither sinking nor rising-at a depth where the density of the surrounding seawater is 1028 { m kg}/{ m m}^{3}? Neglect the mass of the bladder gas.
Express your answer to two significant figures and include the appropriate units.

Explanation / Answer

To be neutrally buoyant, the net force on the fish has to be zero.

Fnet = 0

The only two forces acting on the fish are gravity and buoyancy, so it follows:

Fnet = Fgravity + Fbuoyancy = 0

Fgravity = - Fbuoyancy

mfish g = - water g (Vfish + Vgas)

mfish = - water (Vfish + Vgas)

mfish / water = - Vfish - Vgas

Vgas = mfish / water - Vfish

We know the mass of the fish and the density of the water. We need to know the volume of the fish to solve the problem. To do this we use the definition of density:

DENSITY = MASS / VOLUME

fish = mfish / Vfish

Vfish = mfish / fish

Plugging that it to the above equation yields:

Vgas = - mfish / water + mfish / fish

Vgas = - 10.4kg / (1028 kg/m3) + 10.4kg / (1070 kg/m3) = 0.0003×10-4 m3

          = 0.3 mL

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