Buoyancy and Archimedes' Principle

Links and useful resources

Lightning Round Questions

What shape do Kepler's laws specify for planetary orbits?
What is the formula for calculating work?
Why is the idea of a spring constant important for weighing things?
How do you make a new instance of a class in typescript?
Draw a circuit that uses a pull-up resistor to determine whether a switch is closed or open.
What is the ideal gas law?

gr7: [lightning:: 5]
gr10: [lightning::2]

Concept summary and connections

Archimedes' Principle
Water pressure and depth
Pascal's principle: a change in pressure of an enclosed incompressible fluid is conveyed undiminished to every part of the fluid and to the surfaces of its container.
Pressure potential energy
steady-state flow
streamlines
Bernoulli's equation

Archimedes' principle: An object that displaces a fluid will experience a buoyant force equal to the weight of the displaced fluid.

Weight implies that this is in a gravitational field (or other directional acceleration)
buoyant force means a force that pushes the object upward against the acceleration of gravity
The force always exists, though it may not be enough to cause the object to actually float!

Pascal's Principle

When an incompressible fluid is pressurized inside a vessel, that pressure is immediately transferred through the fluid and to all of the containing surfaces of the vessel. There is no directionality or variation to it. You could also say that an externally applied pressure is exerted everywhere throughout the liquid. Note that this is different from the pressure gradient caused by gravity!

Pressure gradients

When you have a fluid with some depth, the stuff on the bottom has to hold the weight of everything above it. In fact, at every level, the fluid at that level is holding up the weight of all of the fluid above that level. Since fluids exert force through pressure, that means that the pressure in a fluid increases gradually with depth.

The shape of the container makes absolutely no difference to this! Wide parts, narrow parts, tubes that follow winding paths, none of that matters, because of Pascal's principle. The pressure at a given level of depth is the same no matter what , as long as the fluid is connected.

Pressure, work, and energy

We talked about force as it relates to work and energy, and pressure has some similar traits:

work = pressure \times volume

In this equation, you're doing work equal to the volume of fluid you've moved times the pressure you're working against. Likewise, a moving volume of fluid at a given pressure can do work.

This is how hydroelectric dams work: They create a pressure gradient by making a deep pool of water, then they extract work from that pressure at the bottom of the pool by allowing it to jet out through a turbine. The pressure potential energy does work, and is converted in to electrical energy by the turbine.

Flow

When fluid moves, it's called flow. When the flow is continuous and steady, and moves through a stationary environment, it's called steady-state flow. When you watch steady-state flow, you can't detect the passage of time - it looks exactly the same as if the fluid was still. Remember: a fluid is a uniform substance, so we aren't talking about bubbles and twigs in a stream, and if the flow is steady-state, there are no eddies or turbulent spots to reveal what's happening!

If you were to somehow mark a particle of fluid and trace its path as it flows, you would be making a streamline (the path taken by a tiny portio nof fluid in the flow).

In steady state flow, a drop's ordered energy remains constant as it travels along its streamline. Since all drops in a fluid are identical, the ordered energy per drop along a streamline is constant.

Bernoulli's Equation

Ordered energy can take three forms for a drop of fluid: Kinetic energy, pressure potential energy, and gravitational potential energy. The total ordered energy is thus:

\begin{aligned} ordered energy & = pressure potential energy + kinetic energy + gravitational potential energy \\ = pressure \cdot volume + \frac{1}{2} density \cdot volume \cdot {speed}^{2} + m g h \end{aligned}

For the gravitational potential energy, notice that $m = ρ \cdot volume$ , so we can change the equation to be:

ordered energy = pressure \cdot volume + \frac{1}{2} density \cdot volume \cdot {speed}^{2} + density \cdot volume \cdot accel_gravity \cdot height

Dividing both sides by volume gives Bernoulli's Equation:

\begin{aligned} \frac{ordered energy}{volume} & = pressure + \frac{1}{2} density \cdot {speed}^{2} + density \cdot accel_gravity \cdot height \\ = P + \frac{1}{2} ρ v^{2} + ρ g h \\ where \\ ρ & = density \\ P & = pressure \\ v & = velocity \\ g & = acceleration due to gravity \\ h & = height \end{aligned}

This value is constant along a streamline, and it assumes there are no friction-like effects that would remove energy from the system.

Media resources

Homework

#hw (physics) Why do bottles for carbonated drinks always have small caps? 2025-03-07