Aims

- Newton's Laws and Stable Equilibria

- Drag Forces

- Bernoulli's Equation - Lift and Magnus Eect

### 1 Newton's Laws, Weight and Velocity Profile

What are Newton's Laws? How might they be relevant here?

1 - still or constant velocity if no net force

2 - acceleration proportional to net force

3 - equal and opposite reaction of same type

Here the ball hangs in equilibrium - weight and drag must balance.

How does weight vary with height?

Weight essentially constant over the range of the room compared to the radius of the Earth (if they are familiar with binomial expansions could show this mathematically by expanding F = - GMm/(r+h)^2 ).

How would we expect the velocity to vary with distance? Can we draw a graph of this (i.e. velocity on y as function of height on x)?

Well collimated at exit - velocity gradient 0 - can show this using streamers. Falls to 0 at large distance - wouldn't feel effect of fan far away. Doesn't keep going negative - no bulk downward motion above the fan caused (there may be turbulence or circulation at the sides). Smoothly interpolate. Full solution very complicated.

### 2 Drag Forces

What factors will affect the drag?

Relative velocity of wind/object (you feel a force from the air if you run as well as on a windy day, easy to run with wind than into it), size of object, shape (streamlined), also properties of fluid - density, viscosity

In fact the functional form is as follows: At high velocities, inertia effects dominate with constant drag coeffcient:

F = Cpd^2v^2

At low velocities, viscosity dominates. Get Stokes' formula (most likely to have seen this before with Millikan Oil Drop:

F = 6*pi*eta*rv

(Result of drag coeffcient being inverse in velocity i.e.

C = eta/(pdv) )

Note in each case - increasing function of velocity.

Therefore, what would you do if you were building something?

Engineers have to design things to minimise drag by size of area, roughness, stability, smoothness of flow to stop vortices (streamlined tail). Very complicated - get things like vortex shedding which leads to vibration (sounds of wind). Also need to consider laminar and turbulent flow - rough surfaces can actually be better eg golf balls!

Try the "plane wing" (in reality a Pringles tube wrapped in card that you can stick your hand into) in the stream - which way round is it easiest to hold (ignoring the fact it wants to turn a lot) and why? What height is easiest? How does this change when fan turned down?

Should be easier to hold pointing into fan as lower area, more streamlined.

Return to our graph from earlier. Let's assume Stoke's law (where F proportional to v) which means we can relabel y as force. Given weight constant, what should line for weight look like? Where should it be?

Needs to be somewhere between maximum and minimum force such that it crosses the force from the air so we can have equilibrium.

### 3 Stability

Have a closer look at the graph. When the ball moves above the equilibrium, which is greater and thus what happens? What about the other way?

Up - weight greater - falls. Down - drag greater - rises.

This is an example of being stable, when we move away we return. Also described as negative feedback. What other examples of stability are in this problem? Hint: look for other directions that things can move (ball as a whole has 3 directions it can move, with each point having 3 relative to ball):

- The ball doesn't explode or collapse - e.g. the pressure inside goes up if we try to squeeze it. (Accounts for one degree of freedom - distance from centre of mass.)

- The ball often settles a particular way up with nearish the valve at the bottom. Rotational stability familiar from measuring the centre of mass -

if we move away we get a moment that moves us back. (A second degree of freedom - azimuthal angle from centre of mass.)

- The ball doesn't fall out sideways - due to Bernoulli effect which we will come onto. (Accounts for two degrees of freedom - x and y position of whole ball.)

Other examples across the sciences of feedback/stability:

- Populations - predator increases lead to prey decreases which starves predators.

- Homeostasis - e.g. if glucose levels too high, then insulin reduces them and vice versa for glucagon.

- Chemical equilibrium - in reversible reaction, if we increase products, then backward rate increases and we return - relates to Le Chatelier.

- Bond lengths in atoms - balance of electron energies from attraction to two protons to repulsion of atoms. Can be modelled with Lennard-Jones Potential.

- Balls can sit in valleys but not on top of hills!

- Mass on a spring SHM experiment.

With a good group you could go through stable equilibria being potential minima more - but might need building up idea of force as gradient of potential etc. If they know about Taylor series, you can also discuss all minima being quadratic like SHM.

### 4 Bernoulli's Equation - Lift and Magnus Effect

In steady flow (only) the arrows show both the direction of the fluid at any one point and also where a particle will go over time.

Where is the flow around the ball? Try sketching this on paper and confirming hypothesis with streamers.

Important to visualise streamlines using streamers at variety of places - note speed and direction at various heights; distances from centre; bottom, side and top of ball with varying proximity.

#### Bernoulli's Equation

For a unit volume of incompressible fluid (suprisingly good even for gas), energy conservation gives us (derivation not needed here, we are looking at mathematical reasoning):

P + 1/2pv2 + pgh = constant

What do the terms in this equation mean?

P is pressure, p is the density, v is the velocity, gh is gravitational potential

What happens if the velocity at a given height changes?

Pressure must change to balance.

Thus, why does the ball stay in? Try the ping-pong ball. Why does it not stay in?

Pressure difference between side of ball if off-centre. Pressure difference too small on ping-pong ball.

Shows diagram illustrating how the pathlines/streamlines go around a aerofoil (wing). Where are the lines more spread? What does this mean for velocity? Therefore, how does the pressure vary between sides?

Lines spread below and bunch up above. Mass conservation means that lower velocity below meaning higher pressure - net upwards force - lift. Bird Wings also use similar principles - also have narrow tips to minimise vortices - see cross section comparison.

#### Magnus Effect

Demo - a pair of plastic cups taped back to back. Wrap (chain of) elastic bands around them. Hold off of one finger like slingshot and release.

Have a look at this pair of cups. What does it do immediately after firing?

Cups loop up into air.

Why does this happen? Can we draw diagram of flow lines?

Cups drag air as they rotate- adds to motion on one side and detracts on the other meaning there is a velocity and hence pressure difference.

This is known as the Magnus effect. One example is rotor ships. Perhaps more common is the spin on a ball in sports e.g. football, golf, table tennis.