Archimedes’ Principle

Archimedes’ Principle

Figure 15-9 shows a student in a swimming pool, manipulating a very thin plastic sack (of negligible mass) that is filled with water. She finds that the sack and its contained water are in static equilibrium. tending neither to rise nor to sink. The downward gravitational force F.. on the contained water must be balanced by a net upward force  rom the water surrounding thesack. This net upward force is a buoyant force F b : It exists because the pressure in the surrounding water increases with depth below the surface. Thus, the pressure near the bottom of the sack is greater than the pressure near the top. Then the forces on the sick due to this pressure are greater in magnitude near the bottom of the sack than near the top. Some of the forces are represented in Fig. IS-lOa, where the space  occupied by the sack has been left empty. Note that the force vectors drawn near the bottom of that space (with upward components) have longer lengths than those drawn near the top of the sack (with downward components). If we vectorially add all the forces on the sick from the water, the horizontal components cancel and the .

Fb is shown to the right of the pool in Fig. IS-lOa.) Because the sack of water is in static equilibrium, the magnitude of Fh is equal   to the magnitude mfg of the gravitational force r.on the sack of water: Fb = mfg.(Subscriptfrefers toluic, here the water.) In words, the magnitude of the buoyant  force is equal to the weight of the water in the sack.In Fig. IS-lOb, we have replaced the sack of water with a stone that exactly fills the hole in Fig. 15- lOa. The stone is said to displace the water, meaning that it occupies space that would otherwise be occupied by water. We have changed nothing   bout the shape of the hole, so the forces at the hole’s surface must be the same as when the water-filled sack was in place. Thus, the same upward buoyant force that acted on the water-filled sack now acts on the stone: that is, the magnitude Fh of the buoyant force is equal to mfg, the weight of the water displaced by the stone. Unlike the water-filled sack, the stone is not in static equilibrium. The downward gravitational force ~ on the stone is greater in magnitude than the upward buoyant force, as is shown in the free-body diagram to the right of the. pool in F 10h. The stone thus accelerates downward, sinking to the bottom of the pool. Let us next exactly fill the hole in  with a block of light-weight wood, as in Fig. 15-IOc. Again, nothing has changed about the forces at the hole’surface, so the magnitude Fh of the buoyant force is still equal to mfg, the weight of the displaced water. Like the stone, the block is not in static equilibrium. However, this time the gravitational-force ~ is lesser in magnitude than the buoyant force (as shown to the right of the pool), and so the block accelerates upward, rising to the top surface of the water.Our results with the sack, stone, and block apply to all fluids and are summarized .

When a body is fully or partially submerged in a fluid, a buoyant force Fb from the
surrounding fluid acts on the body. The force is directed upward and has a magnitude
equal to the weight m~ of the fluid that has been displaced by the body

The buoyant force on a body in a fluid has the magnitude

F=MG

III the late evening of August 21, 1986. something (possibly a volcanic tremor) disturbed Cameroon’s Lake Nyos, which has a high concentration of dissolved carbon dioxide. The  Disturbance caused that gas to form bubbles. Being lighter than the surrounding fluid (the water), those bubbles were buoyed to the surface, where they released the carbon dioxide. The gas. being heavier than the surrounding fluid (now the air). rushed down the mountainside like a river, asphyxiating 1700 persons and the scores of animals seen here.

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