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 ay = -g

The Bernoullis: A Family of Geniuses

Jakob BernoulliMathematician

Artist

Johann BernoulliMathematician

Chemist

Daniel BernoulliPhysicist

Mathematician

Daniel Bernoulli (1700-1782)

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)

F1

F2

Δx1

Δx2

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.

F1

F2

Δx1

Δx2

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 105 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 = P1

•½ρv12

•h1 = 0

Point B•Atmospheric pressure = P2

•½ρv22

•h2 = 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 105 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 105 Pa + 2,645 Pa = 101,000 Pa + ½(1,000 kg/m3)v22 + (1,000 kg/m3)(9.8 m/s2)(0.50 m)

v2 ≈ 20 m/s

Bernoulli Whiteboard IIAt a desalinization plant, a large tank of saltwater ( = 1,025 kg/m3) is 25 m in height, and open at the top. A small drain plug with a cross-sectional area of 4.0 10-5 m2 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

P1 = Patm

v1 ≈ 0 m/s

h1 = 25 m

P2 = Patm

v2 = ?

h2 = 5 m

v2 ≈ 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 Patm 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 Pexternal 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