Archimedes’ Principle , Floating , Apparent Weight in a Fluid

 

Figure :

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_g on the contained water must be balanced by a net upward force from the water surrounding the sack.

This net upward force is buoyant force F_b. It exits because the pressure in the surrounding water increases with depth below the surface . Thus, the pressure near the bottom of the sack is greather than pressure near the top, which means the forces on the sack due to this pressure are greater in magnitude near the bottom of the sack than near the top.

Some of the force are represented in figure a

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 sack from the water, the horizontal components cancel and the vertical components add to yield the upward buoyant force F_b on the sack. ( Force F_b is shown to the right of the pool in figure a )

Because the sack of water is in static equilibrium , the magnitude of F_b is equal to the magnitude m_fg of the gravitational force F_g on the sack of water: F_b = m_f g. ( subscript f refers to fluid , here tha water ) . In words, the magnitude of the buoyant force is equal to the weight of the water in sack.

In figure b, we have replaced the sack of water with a stone that exactly fills the hole in figure a. The stone is said to displace the water, meaning that the stone occupies space that would otherwise be occupied by water. We have changed nothing about 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 F_b of the buoyant force is equal to m_fg, 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 F_g on the stone is greather in magnitude that the upward buoyant force, as is shown in the free-body diagram in figure b. The stone thus accelerates downward, singking to the bottom of the pool.

Lets us next exactly fill the hole in figure a with a block of lightweight wood, as in figure c. Again , nothing has changed about the forces at the hole’s surface , so the magnitude F_b of the buoyant force is still equal to m_fg, the weight of the displaced water. Like the stone , the block is not in static equilibrium.

However, this time the gravitational force F_g 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 in Archimedes’ principle :

“ When a body is fully or partially submerged in a fluid, abuoyant force F_b from the surrounding fluid acts on the body. The force is directed upward and has a magnitude equal to the weight m_fg of the fluid that has been displaced by the body”

The buoyant force on abody in a fluid has the magnitude

Where m_f is the mass of the fluid that is displaced by the body.

 

Floating

When we release a block of lightweight wood just above the water in a pool, the block moves into the water because the gravitational force on it pulls it down – ward. As the block displaces more and more water, the magnitude F_b of the upward buoyant force acting on it increases. Eventually, F_b is large enough to equal the magnitude F_g of the downward gravitational force on the block, and the block comes to rest. The block is then in static equilibrium and is said to be floating in the water . In general .

“ When a body floats in a fluid, the magnitude F_b of the buoyant force on the body is equal to magnitude F_g of gravitational force on the body”

We can write this statement as

We know that F_b = m_f g . Thus,

“ When a body floats in a fluid, the magnitude F_g of the gravitational force on the body is equal to the weight m_fg of the fluid that has been displaced by the body “

We can write this statement as

In other words, a floating body displaces its own weight of fluid.

 

Apparent Weight in a Fluid

If we place a stone on ascale that is calibrated to measure weight, then the reading on the scale is the stone’s weight. However, if we do this underwater, the upward buoyant force on the stone from the water decreases the reading . That reading is then an apparent weight. In general , an apparent weight is related to the actual weight of a body and the buoyant force on the body by

Which we can write as

If , in some test of strength, you had to lift a heavy stone, you could do it more easily with the stone underwater. Then you applied force would need to exceed only the stone’s apparent weight, not its larger actually weight, because the upward buoyant force would help you lift the stone.

The magnitude of the buoyant force on a floating body is equal to the body’s weight. Equation thus tells us that a floating body has an apparent weight of zero – the body would produce a readingof zero on a scale.

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