Archimedes’ Principle, Principle of Flotation and Equilibrium of Floating bodies

Archimedes’ Principle

When a body is fully or partially immersed in a fluid, it experiences an upthrust which is equal to weight of the fluid displaced by the body.

Suppose a body has weight of W newton in air and volume V. If the body is immersed in a liquid, let the weight of the body in it be W₁. Then

Loss in weight of the body in liquid or buoyancy = W-W₁

According to Archimedes’ principle thrust weight of displaced fluid

upthrust= W-W₁

Experimental Verification

Take a body which is not soluble in a liquid and weight it in air with a spring balance as shown in the figure. Let its weight be W₁. Weigh the body again by immersing the body in the liquid in a beaker without touching the sides and bottom of the beaker. Let the weight be W₂ in the liquid.

The difference in weight = W₁ – W₂

After noting the initial level of a liquid in a measuring cylinder, immerse the body in the liquid and then note the final reading level in it. The difference in liquid levels given in the volume of displaced liquid which is equal to the volume of the body. The weight of displaced liquid = Vρg where ρ is the density of the fluid. It can be observed that

W_{1 -}W₂= V ρ g

In this way, Archimedes’ principle is verified.

Principle of Flotation

If a body is floating in a liquid, its weight W acts vertically downward and the upthrust U due to displaced liquid acts vertically upward. If the weight of the body of volume V is greater than its upthrust, the body will sink in the liquid and will lie at the bottom of the container. So,

\begin{align*} W &> U \\ \text {or,} \: \rho Vg &> V\rho _l g \\ \text {or,} \: \rho &>\rho _l \\\end{align*}

Hence, a body will sink in a liquid as shown in the figure. If the density of the body is greater than the density of the liquid. In case, W=U, the body just sinks and remains inside the liquid with its upper surface near the liquid surface as shown in the figure. So,

\begin{align*} W &=U \\ \text {or,} \: \rho Vg &= V\rho _l g \\ \text {or,} \: \rho &=\rho _l \\\end{align*}

If the weight of the body is smaller than the upthrust, W<U, the body will float on the surface of liquid. So,

\begin{align*} W &< U \\ \text {or,} \: \rho Vg &< V\rho _l g \\ \text {or,} \: \rho &<\rho _l \\\end{align*}

A body will float in a liquid if its density is smaller than the density of the liquid. For example, the density of cork is smaller than that of water, and a cork floats in water. This is shown in figure. So, a floating body displaces the liquid of its own weight. Then,

\begin{align*} mg &= U \\ \text {or,} \: V\rho g &= V_1\rho _1 g \\ \text {or,} \: V\rho &= V_l\rho _1 \\ \text {or,} \: V_1/V &= \rho \rho _l \\ \end{align*}

Where V_l is volume of the displaced liquid = the volume of the body inside the liquid. So, for the floating body,

1. The density of the body is smaller than that of liquid, ρ<ρ_l.
2. The volume of displaced liquid is smaller than the volume of floating body. That is, V₁ < V as the ratio ρ/ρ_l is smaller than one.

Equilibrium of Floating Bodies

When a body is floating in a liquid, its weight acting vertically downward is equal to the upthrust acting vertically upward.

At the equilibrium, the centre of gravity C.G. of the body and centre of buoyancy C.B. of the displaced liquid both lie on the vertical axis.

If the floating body is slightly titled from its equilibrium position, then the C.G and C.B will not lie on the same vertical line, as C.B. shifts away.

The point of intersection of the vertical line passing through C.B. and original vertical line is called the Meta Centre, M.C. of the floating body on the liquid.

Condition that the body regains its equilibrium position or falls in the liquid depends upon the position of M.C. and C.G. of the body. These two possible cases are discussed below.

Three Possible Cases

1. If the C.G. of the body lies below the C.B. of the liquid, which is obtained in heavy bottom body, then the M.C. of the body will lie above the C.G. of the floating body. The body is then acted by a pair of forces, W and U acting C.G. and C.B respectively. These two forces form a couple and this couple tends to bring back the body to its equilibrium position. Thus, the body restores its stable equilibrium.
   The bottom of hydrometer, ships etc are made heavy. That’s why cargo and luggage are stored at the bottom of a ship.
2. If the C.G. of the body lies above the C.B. of the liquid, such as in a heavy top body, and when it is tilted from the equilibrium position, the M.C lies below C.G.
3. The couple formed by two forces W and U acting at C.G. and C.B. rotates more the body and takes it away from the equilibrium position. The body then topples down in the liquid. So stability of the body is lost. So, the people on the boat are not allowed to stand because it is likely to overturn.

Examples of Floatation

1. Ships:
   Though the density of the materials used in ship is greater than the density of water, the structure of the ship is made such that it displaces more volume of water than it does in solid form. The construction of ship makes it floating as it is hollow inside it.
2. Iceberg:
   Density of ice is smaller than the density of water. So, it floats in sea with certain volume out of water level.
3. Submarine:
   Submarine can float as well as sink in sea water. When water is filled up in its water tanks, the submarine becomes heavy and sinks down in sea. When water is forced out of the tanks by high air pressure through valves, the ship can float in sea.
4. Balloons:
   A hydrogen filled balloon is lighter than air and the air forces up it to a height where weight of balloon is equal to the upthrust of air there.