A steel sphere, \(1.5 \mathrm{~mm}\) diameter and of mass \(13.7 \mathrm{mg}\), falls steadily in oil through a vertical distance of \(500 \mathrm{~mm}\) in \(56 \mathrm{~s}\). The oil has a density of \(950 \mathrm{~kg} \cdot \mathrm{m}^{-3}\) and is contained in a drum so large that any wall effects are negligible. What is the viscosity of the oil? Verify any assumptions made.

When an object falls through a fluid, it eventually reaches a constant speed known as terminal velocity. This happens when the forces acting on the object - namely gravitational force, drag force, and buoyant force - balance out. In this exercise, the steel sphere reaches terminal velocity as it falls through oil. The formula for terminal velocity, v, is fairly straightforward:\( v = \frac{\text{distance}}{\text{time}} \)Here, the steel sphere falls a distance of 0.5 meters in 56 seconds, giving us a terminal velocity of approximately 0.00893 m/s.

Buoyant force is the upward force that a fluid exerts on an object immersed in it. This force acts opposite to gravity and is responsible for making objects feel lighter in water. For the steel sphere in this exercise, the buoyant force depends on the volume of the sphere and the density of the oil. Buoyant force is calculated using the formula:\( F_b = \rho_{\text{fluid}} \times V \times g \)where \( \rho_{\text{fluid}} \) is the density of the fluid, V is the volume of the object, and g is the acceleration due to gravity (9.81 m/s\text{^{2}}). By substituting the given values, the buoyant force on the steel sphere is found to be approximately \( 1.647 \times 10^{-5} \text{~N} \).

Stokes Law helps us understand the drag force experienced by small spherical objects moving slowly through a viscous fluid. According to Stokes Law, the drag force, \( F_d \), is given by:\( F_d = 6 \times \text{pi} \times \text{eta} \times r \times v \)Here, \( \text{eta} \) is the fluid’s viscosity, r is the radius of the sphere, and v is the terminal velocity. For this exercise, the problem statement provided all necessary data to apply Stokes Law. Balancing the gravitational force, buoyant force, and drag force, we find the viscosity of the oil to be approximately \( 0.114 \text{~Ns \text{\textperiodcentered} m^{-2}} \). This calculation validates our understanding and assumptions, while demonstrating the practical application of Stokes Law.

Fluid dynamics studies the behavior of liquids and gases in motion. In this exercise, we explore a simple yet insightful fluid dynamics problem where a steel sphere falls through oil. The interplay of different forces - gravity, buoyancy, and drag - highlights fundamental principles in fluid dynamics. An understanding of fluid properties such as density and viscosity is crucial. Calculations involving fluid dynamics help to predict and describe the motion and behavior of objects in fluids. These principles have wide applications, from industrial processes to natural phenomena. By systematically solving such problems, students build a solid foundation in the intricate world of fluid dynamics.