A piston \(113 \mathrm{~mm}\) diameter and \(150 \mathrm{~mm}\) long has a mass of \(9 \mathrm{~kg}\). It is placed in a vertical cylinder \(115 \mathrm{~mm}\) diameter containing oil of dynamic viscosity \(0.12 \mathrm{~Pa} \cdot \mathrm{s}\) and relative density \(0.9\) and it falls under its own weight. Assuming that piston and cylinder are concentric, calculate the time taken for the piston to fall steadily through \(75 \mathrm{~mm}\).

Dynamic viscosity is a measure of a fluid's resistance to flow. It describes how much resistance a fluid offers when subjected to shear stress. Higher viscosity means the fluid is thicker and flows less easily. In our exercise, the oil in the cylinder has a dynamic viscosity of 0.12 Pa·s. This tells us how sticky or thick the oil is, which directly affects how it will slow down the falling piston. For engineers and students, it's crucial to understand how viscosity impacts movement in fluid mechanics problems. The unit for dynamic viscosity in the SI system is Pascal-seconds (Pa·s). When calculating forces in fluid dynamics, remember to incorporate viscosity since it significantly influences the final results.

Gravitational force is the force exerted by gravity on an object. It is calculated as the product of the object's mass and gravitational acceleration. In this exercise, the piston with a mass of 9 kg falls under its weight, experiencing a gravitational pull. We used the formula \( F_g = m \cdot g \) to find it. Substituting in the values, we get \( F_g = 9 \cdot 9.81 = 88.29 \ \text{N} \). This force drives the piston downward, where it is then counteracted by the viscous force of the oil. Understanding gravitational force helps in predicting how objects will move when falling, which is essential in fluid mechanics and other physics-related fields.

A steady-state condition occurs when all forces acting on an object are balanced, resulting in constant velocity. In our problem, the piston reaches steady state when gravitational force is equal to the viscous drag force exerted by the oil. We establish this condition with the equation \( F_g = F_v \). Both forces being equal means there is no acceleration, just consistent falling speed. The concept of steady-state is vital in many engineering and physics scenarios because it simplifies the analysis by indicating a stable situation without acceleration. Understanding this condition helps predict system behavior under long-term operation.

Clearance refers to the gap or spacing between two components. In this exercise, clearance is the space between the piston and the cylinder wall. It's calculated to consider how oil flows in the gap, affecting the viscous force. We use the formula: \( h_c = \frac{0.115 - 0.113}{2} = 0.001 \ \text{m} \). This small gap significantly influences the calculations for viscous drag. In fluid mechanics, precise clearance measurements are crucial because even slight changes can impact the resistance force drastically, affecting the system's performance.

SI unit conversion ensures consistency and accuracy in scientific calculations. Converting measurements to SI units (meters, kilograms, seconds) helps standardize the values across all steps. For example, in our exercise, we converted diameters from millimeters to meters (113 mm to 0.113 m). This makes the subsequent calculations more straightforward and prevents errors. Always make sure to convert units at the beginning to avoid confusion and ensure that all formulas and constants align properly. Using SI units is essential in engineering and science for seamless communication and comparison of data worldwide.