20. Fluid Dynamics and Statics and Bernoulli's Equation

Professor Shankar begins by introducing the topic of fluid dynamics and statics, noting that it's a fundamental subject even for those with high school physics knowledge. He explains that fluids, like water or oil, can be characterized by properties such as density (ρ), and that pressure, which increases with depth in a fluid, is a key concept. The formal definition of pressure is discussed, highlighting that it's a scalar quantity, measured as force per unit area, and denoted in Pascals. The concept of gauge pressure and absolute pressure is also introduced, with examples provided to illustrate these principles.

The paragraph delves into the principle that pressure in a fluid at a given depth is the same in all directions, leading to equilibrium. Using the mental model of an imaginary cylindrical section within the fluid, it's demonstrated that any imbalance in pressure would result in net force, contradicting the equilibrium state. The pressure difference between two depths in a fluid is then calculated, showing that it depends on the density of the fluid, gravitational acceleration, and the height difference between the two points.

This section discusses the concept of atmospheric pressure, which is the pressure exerted by the weight of the air above us. It is explained that although we live at the bottom of a 100-mile-high 'pool' of air, we do not experience the full pressure due to the air passing through our bodies. The relationship between atmospheric pressure, density, and height is given as ρgh, with the standard atmospheric pressure being 10^5 Pascals. An experiment involving a heated and sealed can is used to illustrate changes in pressure.

The applications of the principles of pressure are explored, including the creation of a barometer to measure atmospheric pressure by using the height of a mercury column. The principle behind drinking through a straw is also explained, which involves creating a pressure differential to draw liquid up the straw. Additionally, the concept of a U-tube filled with different fluids is introduced to demonstrate how the difference in fluid heights can be used to determine the relative densities of the fluids.

The hydraulic press, which uses the principle of pressure to amplify force, is described. It's shown that by applying force to a smaller piston in a hydraulic system, a larger force can be exerted on a larger piston due to the conservation of energy and the incompressible nature of the fluid. Archimedes' principle is then introduced, stating that an object submerged in a fluid is buoyed up by a force equal to the weight of the fluid displaced by the object. The principle is applied to explain why objects of different densities float or sink, and the concept of the fraction of an object's volume that will be submerged based on its density relative to the fluid is discussed.

The principles of buoyancy are further elaborated through the example of a boat floating on water. It's explained that a boat made of steel can float due to its hollow design, which displaces a volume of water equal in weight to the boat itself. The critical loading point of a boat, where it is on the verge of sinking, is also discussed, highlighting the balance between the boat's weight, cargo, and the weight of the displaced water. The paragraph concludes with a mention of Bernoulli's Equation, which will be addressed in the next topic, and its significance in considering fluids in motion.

The 'equation of continuity' is introduced, which relates the flow rate of an incompressible fluid through different sections of a pipe. It's shown that the product of the cross-sectional area and the velocity of the fluid at any point must be constant, leading to the conclusion that as the pipe's cross-sectional area decreases, the fluid's velocity must increase to maintain continuity. The implications of this principle for fluid dynamics, such as the behavior of water in pipes of varying sizes, are discussed.

Bernoulli's equation is derived and explained in the context of fluid dynamics. The equation, which states that the sum of pressure, kinetic energy, and potential energy per unit volume remains constant along a streamline, is applied to various scenarios. Examples include the behavior of a spinning baseball, the lift generated by an airplane wing, and the functioning of an atomizer. The principle that an increase in the speed of a fluid results in a decrease in pressure is highlighted, and the qualitative and quantitative aspects of Bernoulli's equation are discussed.

The final paragraph explores practical applications of Bernoulli's principle, such as measuring the speed of an airplane or the flow rate of a liquid in a pipe using a device called a Venturi meter. The principle that a decrease in pressure corresponds to an increase in fluid velocity is utilized to determine these velocities. The Venturi meter works by creating a constriction in the pipe, which increases the fluid velocity and decreases the pressure, measurable by the difference in height of a fluid in a secondary tube. This principle is also used to explain how the height of fluid in an open tube adjusts when the tube is opened or closed to the atmosphere.