Curveballs: Explained! Since the ball is spinning forward, the air above the ball is slowed down significantly (by friction), while the air below the ball flows smoothly under it. This creates slower, higher-pressure air above! The pressure differential causes the ball to drop faster than a y = -g

The Bernoullis: A Family of Geniuses Jakob Bernoulli Mathematician Artist Johann Bernoulli Mathematician Chemist Daniel Bernoulli Physicist Mathematician

Daniel Bernoulli ( ) Became very interested in determining a way to measure blood pressure by using physical principles. His quest led him to discover the most fundamental principal of fluid dynamics.

For any individual object, the Law of Conservation of Energy always applies. This means that in the absence of elastic energy or changes in thermal energy, This applies between any two points along the object’s motion, energy will be conserved.

Daniel Bernoulli was the first person to make the following conceptual leap. And so, Bernoulli went on to construct the Law of Conservation of Energy per unit volume. If the Law of Conservation of Energy applies to one particle, then it must also apply to groups of particles! In fact, any chunk of fluid will obey the LoCoE! (jealous)

F1F1 F2F2 Δx 1 Δx 2 This applies to any given chunk of fluid that passes through this section of pipe negative work done by the external force here!

Law of Conservation of Energy for a Fluid Daniel Bernoulli essentially divided this entire equation by volume.

F1F1 F2F2 Δx 1 Δx 2 V V But, what is FΔx/V?

Whiteboard Challenge! What is ? Use a unit analysis or geometric analysis if you are stuck!

Bernoulli’s Equation!!!! Otherwise written as External pressure exerted on point 1 External pressure exerted on point 2

Bernoulli: Example Problem An underground irrigation system uses a subterranean pump to provide water to a field of pickle plants. The pump provides an underground pressure of 3 x 10 5 Pa, and the flow speed of the underground water is 2.3 m/s. What is the speed of the water as it exits the pipe at ground level?

Point A Pressure from the pump = P 1 ½ρv 1 2 h 1 = 0 Point B Atmospheric pressure = P 2 ½ρv 2 2 h 2 = 0.50 m

An underground irrigation system uses a subterranean pump to provide water to a field of pickle plants. The pump provides an underground pressure of 3 x 10 5 Pa, and the flow speed of the underground water is 2.3 m/s. What is the speed of the water as it exits the pipe at ground level? Go for it!!

3 x 10 5 Pa + 2,645 Pa = 101,000 Pa + ½(1,000 kg/m 3 )v (1,000 kg/m 3 )(9.8 m/s 2 )(0.50 m) v 2 ≈ 20 m/s

Bernoulli Whiteboard II At a desalinization plant, a large tank of saltwater (  = 1,025 kg/m 3 ) is 25 m in height, and open at the top. A small drain plug with a cross-sectional area of 4.0  m 2 is 5.0 m from the floor. a) Calculate the speed of the saltwater as it leaves the hole on the side of the tank when the hole is unplugged. b) If this were freshwater instead of saltwater, how would it affect your answer?

Solution P 1 = P atm v 1 ≈ 0 m/s h 1 = 25 m P 2 = P atm v 2 = ? h 2 = 5 m v 2 ≈ 20 m/s!

General Strategy for Bernoulli’s Equation Choose two points in the path of the fluid, and stick with those two points If one of the points is open to the atmosphere, use P atm as the external pressure. Feel free to set the zero height level to be where it is most convenient. Density of the fluid will not divide out of the equation – don’t forget the P external terms!

Bernoulli Party! Please use the remainder of the class to work on the following questions with your groups. – Anything that is not finished in class will be completed as homework. Your quiz tomorrow will involve similar problems, so use your time in class wisely