A solid uniform cylinder of length \(150 \mathrm{~mm}\) and diameter \(75 \mathrm{~mm}\) is to float upright in water. Between what limits must its mass be?

Archimedes' Principle is a fundamental concept in fluid mechanics. It states that an object immersed in a fluid experiences a buoyant force equal to the weight of the fluid displaced by the object. In simpler terms, this force helps objects float. For example, when a cylinder is placed in water, it displaces a certain amount of water. The weight of this displaced water is what keeps the cylinder afloat. This principle is crucial for understanding why objects sink or float based on their density compared to the fluid they are in.

The buoyant force is the upward force exerted by a fluid on an immersed object. This force can be calculated using Archimedes' Principle. The formula for buoyant force in this scenario is: \[ \text{Buoyant force} = \rho \times V \times g \] where \( \rho \) is the density of the fluid, \( V \) is the volume of the displaced fluid, and \( g \) is the acceleration due to gravity. For a cylinder to float, the buoyant force needs to balance the gravitational force acting on the cylinder. This means the weight of the displaced water must be equal to or greater than the weight of the cylinder. In the given exercise, the cylinder's weight must be within the range that allows it to displace enough water to stay afloat.

Calculating the volume of a cylinder is essential in determining the buoyant force. The volume \( V \) of a cylinder can be calculated using the formula: \[ V = \frac{\text{π} d^2 l}{4} \] where \( d \) is the diameter and \( l \) is the length of the cylinder. In the exercise, the given diameter is 0.075 meters and the length is 0.15 meters. Substituting these values into the formula, we get: \[ V ≈ 6.627 \times 10^{-4} \text{ m}^3 \] This volume is then used to calculate the buoyant force, which helps determine the mass limit for the cylinder to float.

Fluid mechanics is the study of fluids (liquids and gases) and the forces acting on them. It includes concepts such as fluid statics, dynamics, and kinematics. Understanding fluid mechanics is essential for problems involving buoyancy, like the one in the exercise. When dealing with buoyancy calculations, knowledge of fluid properties such as density is crucial. For water, the density is typically \( 1000 \text{ kg/m}^3 \). This property is used in calculating the buoyant force experienced by objects submerged in the fluid. The principles of fluid mechanics help us analyze and predict the behavior of objects in various fluids, guiding us in real-world applications like ship design and underwater exploration.