Section 7.5
Power Required to Remain Afloat
7.4
Archimedes’ Principle Archimedes’ principle states that a body partially or wholly submerged in a fluid is buoyed upward by a force that is equal in magnitude to the weight of the displaced fluid. The derivation of this principle is found in basic physics texts. We will now use Archimedes’ principle to calculate the power required to remain afloat in water and to study the buoyancy of fish.
7.5
Power Required to Remain Afloat
Whether an animal sinks or floats in water depends on its density. If its density is greater than that of water, the animal must perform work in order not to sink. We will calculate the power P required for an animal of volume V and density ρ to float with a fraction f of its volume submerged. This problem is similar to the hovering flight we discussed in Chapter 6, but our approach to the problem will be different.
Because a fraction f of the animal is submerged, the animal is buoyed up by a force FB given by FB gf Vρw
(7.9)
where ρw is the density of water. The force FB is simply the weight of the displaced water.
The net downward force FB on the animal is the difference between its weight gVρ and the buoyant force; that is, FD gVρ − gVfρw gV (ρ − fρw)
(7.10)
To keep itself floating, the animal must produce an upward force equal to FD. This force can be produced by pushing the limbs downward against the water. This motion accelerates the water downward and results in the upward reaction force that supports the animal.
If the area of the moving limbs is A and the final velocity of the accelerated water is v, the mass of water accelerated per unit time in the treading motion is given by (see Exercise 7-1) m Avρw
(7.11)
Because the water is initially stationary, the amount of momentum imparted to the water each second is mv. (Remember that here m is the mass accelerated per second.)
Momentum given to the water per second mv
7.4
Archimedes’ Principle Archimedes’ principle states that a body partially or wholly submerged in a fluid is buoyed upward by a force that is equal in magnitude to the weight of the displaced fluid. The derivation of this principle is found in basic physics texts. We will now use Archimedes’ principle to calculate the power required to remain afloat in water and to study the buoyancy of fish.
7.5
Power Required to Remain Afloat
Whether an animal sinks or floats in water depends on its density. If its density is greater than that of water, the animal must perform work in order not to sink. We will calculate the power P required for an animal of volume V and density ρ to float with a fraction f of its volume submerged. This problem is similar to the hovering flight we discussed in Chapter 6, but our approach to the problem will be different.
Because a fraction f of the animal is submerged, the animal is buoyed up by a force FB given by FB gf Vρw
(7.9)
where ρw is the density of water. The force FB is simply the weight of the displaced water.
The net downward force FB on the animal is the difference between its weight gVρ and the buoyant force; that is, FD gVρ − gVfρw gV (ρ − fρw)
(7.10)
To keep itself floating, the animal must produce an upward force equal to FD. This force can be produced by pushing the limbs downward against the water. This motion accelerates the water downward and results in the upward reaction force that supports the animal.
If the area of the moving limbs is A and the final velocity of the accelerated water is v, the mass of water accelerated per unit time in the treading motion is given by (see Exercise 7-1) m Avρw
(7.11)
Because the water is initially stationary, the amount of momentum imparted to the water each second is mv. (Remember that here m is the mass accelerated per second.)
Momentum given to the water per second mv