Full explanation on archimedes principle

Question

and law of floatation with calculation examples

Answer 1

Archimedes' Principle

Definition: Archimedes' Principle states that any object wholly or partially submerged in a fluid experiences a buoyant force equal to the weight of the fluid displaced by the object. This principle explains why objects float or sink in fluids.

Key Points:

1. Buoyant Force (B): The upward force exerted by the fluid on the submerged object.
2. Displacement: When an object is submerged, it pushes out a volume of fluid equal to the volume of the part of the object that is submerged.
3. Weight of Displaced Fluid: The weight of the fluid that is displaced can be calculated using the formula: $$\text{Weight of displaced fluid}=\text{Volume of displaced fluid}\times \text{Density of fluid}\times g$$ where $g$ is the acceleration due to gravity.

Law of Floatation

Definition: The Law of Floatation states that a floating object displaces its own weight of the fluid. For an object to float, the buoyant force acting on it must equal the weight of the object.

Key Points:

1. For an object to float, the weight of the object (W) must be equal to the buoyant force (B). $$B=W$$
2. If the density of the object is less than the density of the fluid, it will float. If it is greater, the object will sink.

Example Calculations

Example 1: A Solid Cube

Let's consider a cube of wood with a side length of 0.5 m and density of 600 kg/m³ that is submerged in water (density of water = 1000 kg/m³).

1. Calculate the volume of the cube: $$V={\text{side}}^3=(0.5\text{ m}{)}^3=0.125{\text{ m}}^3$$

2. Calculate the weight of the cube: $$W=\text{mass}\times g=\text{density}\times V\times g=600{\text{ kg/m}}^3\times 0.125{\text{ m}}^3\times 9.81{\text{ m/s}}^2$$ $$W=600\times 0.125\times 9.81=735.75\text{ N}$$

3. Calculate the buoyant force (weight of displaced water): $$B=\text{Volume of displaced water}\times \text{Density of water}\times g=V\times \text{Density}\times g$$ $$B=0.125{\text{ m}}^3\times 1000{\text{ kg/m}}^3\times 9.81{\text{ m/s}}^2=1226.25\text{ N}$$

4. Observe the behavior:

   • Buoyant force $B=1226.25\text{ N}$
   • Weight of the cube $W=735.75\text{ N}$

Since $B>W$, the cube floats.

Example 2: A Metal Block

Consider a block of metal with a volume of $0.03,{\text{m}}^3$ and density $7800,{\text{kg/m}}^3$.

1. Calculate the weight of the metal block: $$\text{Weight (W)}=\text{Density}\times \text{Volume}\times g=7800{\text{ kg/m}}^3\times 0.03{\text{ m}}^3\times 9.81{\text{ m/s}}^2$$ $$W=7800\times 0.03\times 9.81=2293.74\text{ N}$$

2. Calculate the buoyant force when submerged:

   • Since the block is submerged, it displaces a volume of water equal to its own volume: $$B=\text{Volume of displaced water}\times \text{Density of water}\times g=0.03{\text{ m}}^3\times 1000{\text{ kg/m}}^3\times 9.81{\text{ m/s}}^2$$ $$B=0.03\times 1000\times 9.81=294.3\text{ N}$$

3. Behavior:

   • Buoyant force $B=294.3\text{ N}$
   • Weight of the block $W=2293.74\text{ N}$

Since $B<W$, the metal block sinks.

Conclusion

Archimedes' Principle and the Law of Floatation are fundamental concepts in fluid mechanics that explain the buoyancy of objects in fluids. By applying these principles through various calculations, one can predict whether an object will float or sink when placed in a fluid.