### DYNAMICS MOVIES: Hugh Hunt, Cambridge University Engineering Department

(have a look at other fun dynamics videos at my Youtube site and Tweet me at @hughhunt )

( Gyroscopes and Boomerangs: go to Hugh Hunt's gyro and boomerang page and for other stuff: go to Hugh Hunt's home page )

This site contains several movies that illustrate the dynamics of spin. Dynamics is a boring subject without spin: a translating body does nothing interesting, and an ideal particle behaves much as you would expect. Some of the phenomena illustrated below are very complicated to understand, but very easy to observe. I hope these videos will help to open a few eyes to the wonders of spin.
If you have trouble viewing the videos then go to testpage or download RealPlayer.V.10

## Bouncing Balls

A superball is a solid rubber spherical ball. It is elastic and close to lossless. It also grips any surface so contacts can be assumed no-slip. All the simulations below are based on three assumptions: (1) conservation of energy; (2) conservation of moment of momentum about any point of impact; (3) no slip. The model used for impact is the elastic particle model. It assumes that the point of contact stores and releases elastic energy in zero time.

Ball bouncing under a table

(MPG) or (CIN) or (AVI) and a simulation (MPG) or (CIN) or (AVI)

Note that the ball comes back from where it came
Note the spin of the ball and that it exits without spin in the direction that it entered. Isn't it extraordinary that the ball returns almost exactly along its entry path? A light beam certainly wouldn't do this, so strengthening the case for exercising caution when referring to the "particle theory of light". The particle theory of light assumes that the particle has mass but no moment of inertia and that it has no physical size. I've never seen a ball in the playground that fits this description!

Lampard's disallowed goal in the 2010 World Cup

Lambard's non-goal popped back to the keeper because of spin - to find out why, click here .

Downball - Simulation of a ball bouncing against a wall, hitting the wall first

(MPG) or (CIN) or (AVI)
Note that the exit angle is not equal to the entry angle, and note the spin. It is for this reason that I object to the "particle theory of light" being taught in schools without some kind of clear indication that superballs (or any ball for that matter) do not behave as particles. How can kids be expected to understand the particle theory of light if experiments that they can easily perform with "particles" in the playground demonstrate that particles don't behave as light does?

Downball - Simulation of a ball bouncing against a wall, hitting the floor first

(MPG) or (CIN) or (AVI)
Note now that the exit angle is yet again different from the entry angle. If the particle were to behave as if it were a beam of light then the exit path would be parallel to the entry path. But then ball games in the playground wouldn't be nearly as much fun without spin!

Ball rolling up to a wall - normal incidence
First assume perfect friction conditions with the wall, allowing the rolling ball to grip momentarily onto the wall, lifting itself up into the air.
(MPG) or (CIN) or (AVI) and a simulation (MPG) or (CIN) or (AVI)

Note that the ball doesn't just bounce back - it jumps in the air DRY WALL
The upward force that causes the ball to jump also causes the spin direction to reverse. This reversal of spin is exactly what is needed for the ball ultimately to begin rolling back the other way.

Now if we assume that the wall is slippery, then the ball cannot be lifted up into the air. As a result there is no force from the wall to change the direction of spin so the ball begins its return journey spinning the wrong way. Friction with the floor gradually acts to reverse the spin direction, but this friction force also slows the ball down and in the process the ball is almost brought to a complete halt.
(AVI) ... but now with a slippery wall the ball doesn't jump up OILY WALL
This is completely counterintuitive - who'd have thought that lubricating the wall would cause the ball to bounce back more slowly?

Ball rolling up to a wall - incidence 45 degrees

(MPG) or (CIN) or (AVI) - this is a top view.
Note that the ball does not reflect at 45 degrees. It is airborne and then it changes direction after the first bounce. Note also the change in spin as it bounces.

## Ball bouncing on a bat

A ball bouncing on a sinusoidally-oscillating bat is possibly the simplest impact problem that can be modelled and it demonstrates the problems inherent in modelling impacts.

The videos below give three examples, two steady-state limit cycles and one example of chaotic motion. The limit cycles are very special circumstances that arise given particular intial conditions. Chaos is the usual situation. In any case, the simulations are very-much idealised and any experiemnt will do diferent things yet again. This is the nature of non-linear systems and chaotic motion. The world is full of non-linear phenomena so how can we expect to model it accurately? We can only ever hope to scratch the surface.
(the AVI videos show the full development of steady state whereas the mini animations show a small steady-state snapshot of the motion. The JPG links show the motion graphically)

(AVI) - ball on bat in a simple limit cycle - note the ball hits the bat once per cycle (JPG)

(AVI) - ball on bat in a complex limit cycle - the bat oscillates at twice the ball bounce rate and the ball height alternates (JPG)

(AVI) - ball on bat in chaotic motion - this is the usual condition. (JPG)

Modelling is difficult for three reasons:
1 - The impact is hard to model. Most people use "coefficient of restitution" and this is a simplistic approach. But there are not many alternatives! I have used a coefficient of restitution e=0.9 for all the animations here. The biggest concern is that impact is assumed to be instantaneous - which it isn't. A more-realsitic model has to include a spring at the contact interface and this spring may itself be non-linear and dissipative.
2 - Even for a simple ball with a single contact point bouncing on a sinusoidally-moving table the problem is nonlinear and chaotic solutions result. Solving such systems is not for the feint-hearted.
3 - Being a non-linear and chaotic system it is very sensitive to the method of modelling and also sensitive to initial conditions.
Then when it comes to modelling more realistic problems (for instance an object bouncing in the back of a truck on a bumpy road) it is necessary to include multiple contacts and these add significant complexities and uncertainties.

The Matlab code used to generate this bat-on-ball motion can be found here . This code comes with no guarantees.

## Impact with two masses

Here is a potentially more complicated problem, but it is not driven by a steady input so it doesn't exhibit the classic chaotic behaviours seen above with the bat and ball. A heavy mass (2000kg) is being dropped from a height (20mm) onto a resilient platform (500kg). The mass rebounds off the platform and the platform oscillates. The mass then strikes the platform and second time, and then a third until it finally settles down. Unlike the bat-on-ball problem the contct times are finite and in computing the motion it is necessary to separate out motion when the springs are in contact and when there is loss of contact.

(AVI) - two-mass impact problem (JPG)

The Matlab code used to generate the motion shown can be found here . This code comes with no guarantees.

## Chaotic Pendulum

The ball bouncing on a sinusoidally-oscillating bat is an example of chaotic motion. The double pendulum is another. So very simple (just two degrees of freedom) and yet its motion is chaotic - that is to say its motion is highly unpredictable and very-sensitively dependent on the exact initial conditions.

(AVI) - chaotic motion of a double pendulum - the motion is unpredictable (JPG)

The obvious question to ask is "how can the motion be modelled yet at the same time be unpredictable"? It is true that I have made a mathematical model and I have simulated the motion to produce the animation here. But if an experiment were performed the observed motion would not be the same as the prediction here. This is because the initial conditions used in the simulation cannot be exactly the same as those in an experiment. Also the presence of small deviations from "ideal behaviour", for instance friction or perhaps flexibility of the bearings, or lack of rigidity of the pendulum rod etc. etc. will lead to massive deviations from the model predictions.

The Matlab code used to generate this chaotic doublependulum motion can be found here . This code comes with no guarantees.

## Falling Toast

 Toast always seems to land buttered-side down when you drop it. This is because when it falls off the edge of a table it starts to rotate. You can do good experiments with a small hard-back book (use a rubber band to hold the pages closed). The motion can be analysed using Newton's Laws of Motion, but it is not as simple as analysts such as Matthews make out. Bacon et al. do it better. There are three phases, "no slip", "slip" and "free fall". The first thing to do is to hold the book (or toast) over the edge of a table so that it will just balance. Then push it just 1mm further and it will start to fall. At first the angular velocity of the book increases gradually during the "no slip" phase until it reaches an angle of around 20o - 30o (depending on friction). At this point friction is insufficient to prevent slip, but the book does not lose contact. During the "slip" phase (you can hear the book slip) the forces acting on the book are more-widely separated so the angular acceleration is more rapid. This means that it is during this slipping phase that the book gains most of its angular velocity. Eventually the angular motion causes the book to lose contact with the table and it then enters the "free fall" stage. The angular velocity is constant but simple projectile motion requires that the book accelerate downwards at a rate of g = 9.81ms-1 and in the time it takes the book to turn through 180o it has fallen a distance of about 70cm. This is the height of a typical table. What is remarkable is that this conclusion is not very sensitive to the exact starting position of the book (1mm, 2mm - doesn't matter much) or the friction coefficient (anything between µ = 0.1 and 0.5 will give roughly the same answers. So it is no surprise that toast often lands upside down. For a fuller explanation see here . The Matlab code used to generate this falling toast motion can be found here . This code comes with no guarantees. Download a copy of the movie at (MPG) , (WMV) or (AVI)

## Pencils

(these make excellent exam questions)

(MPG) or (CIN) or (AVI) - Pencil falling onto the edge of a table - note that the pencil loses some of its energy on collision

(MPG) or (CIN) or (AVI) - Pencil hit off the edge of a table - just hits

(MPG) or (CIN) or (AVI) - Pencil hit off the edge of a table - just misses

## Ball rolling on a rotating table

These movies show the classic and counter-intuitive motion of a ball rolling without slip on a horizontal rotating turntable. Note that the ball moves on a space-fixed circle whose centre and radius are determined solely by initial conditions. The period of motion of the ball on its circle is 7/2 that of the table (5/2 for a hollow sphere, eg a ping-pong ball). These results are independent of the size and mass of the ball. If the table is sloping then the centre of the ball convects with a constant velocity perpendicular to the slope as shown in the last of the movies.

(MPG) or (CIN) or (AVI)
Ball moving on a circle centred on the centre of the table

(MPG) or (CIN) or (AVI)
Ball moving on a small circle centred some way off the table centre>

(MPG) or (CIN) or (AVI)
Ball stationary (ie a circle of zero radius)

(MPG) or (CIN) or (AVI)
Ball rolling on a larger circle passing through the table centre

(MPG) or (CIN) or (AVI) - Ball rolling on a sloping table
Because this is a big video you may have trouble downloading it. It has to be big in order to show the ball rolling on the sloping table before being spun up, then to show the spinup, then showing the remarkable convection at 90 degrees to the expected direction. Here is a smaller version: (AVI)
Provided that you trust that the slope of the table in the following two clips is the same, then these files are smaller and just as remarkable.
Ball on sloping table, table not spinning (AVI) or (MPG) or (CIN)

## Ball rolling inside a cylinder

This shows the beautiful motion of a ball rolling without slip inside a fixed cylindrical shell. The period of up-and-down motion of the ball is sqrt(7/2) that of the orbiting period, independent of the size and mass of the ball. The amplitude of up-and-down motion depends on initial conditions and, as the video shows, it is possible to cause the ball to leave the cylinder. The best kind of ball to use is a rubber-coated steel ball (ie a mouse ball).

(MPG) or (CIN) or (AVI) - Ball moving inside a circular cylinder

## Wobblerings

These videos show the motion of a ring wobbling down a rod. These derive from the study of a toy known variously as a gyro ring (gyroring gyro-ring), a wobble ring or a jitter max. There are two motions, one with a single contact point and another with two contact points. The two-contact-point motion is exhibited in the toys, but the single-contact-point motion is quite beautiful and counter-intuitive- only observed with thin rings.

(MPG) or (CIN) or (AVI) - Single-contact-point motion - note the beautiful up-and-down bouncing motion

(MPG) or (CIN) or (AVI) - Double-contact-point motion - note that the motion is very fast compared with single-point motion

(MPG) or (CIN) or (AVI)
Single-contact motion on a horizontal rod - motion on a horizontal rod removes the effect of gravitationally-induced slip convection

(MPG) or (CIN) or (AVI) - Double-contact motion on a horizontal rod

(MPG) or (CIN) or (AVI) - Two-contact-point motion for a ring moving down a rod

(MPG) or (CIN) or (AVI) - The same motion, but viewed from above - note that it looks as if the ring makes a "flip", but this is an illusion resulting from the camera angle. The motion is in fact steady

(MPG) or (CIN) or (AVI)
The toy, showing normal two-contact-point motion of five rings

(MPG) or (CIN) or (AVI) - A closeup slow-motion of the toy

## Stability of a rigid body spining freely in space

(MPG) or (CIN) or (AVI) - The classic unsteady motion of a rigid body about the intermediate moment of inertia

(for a geometric interpretation of this motion involving momental and energetic ellipsoids, see momental ellipsoids for a tumbling rigid-body)

The motion is governed by conservation laws (kinetic energy and moment of momentum). These laws are simple but solution of the equations of motion (Euler's equations) is tricky, involving Jacobian Elliptic Functions. Matlab has been used to procuce the following simulated motions:

(MPG) or (CIN) or (AVI) - Spin about the "A" axis( stable)

(MPG) or (CIN) or (AVI) - Spin about the "B" axis (unstable)

(MPG) or (CIN) or (AVI) - Spin about the "C" axis( stable)

## Swingball (Totem tennis)

The motion of the ball is governed by two conservation laws:

• Conservation of Energy
• Conservation of Moment of Momentum (Angular Momentum, if you prefer) It is easy to write down the equations of motion in terms of energy and moment-of-momentum, but solution of the equations is problematic. Here are two video simulations of the motion of the ball given two different initial conditions.

(MPG) or (CIN) or (AVI) - a "gentle" hit which doesn't go over the head of your opponent

(MPG) or (CIN) or (AVI) - a "smash" which goes over the head of your opponent - this is aggressive play!

Equations of Motion - are given here. The equations are simple, but the solution is tricky.

## Rattlebacks

Mysterious rigid-body motions are exhibited by boat-shaped object spun on a flat surface. Smooth river-worn stones also show this motion. A given object will have a preferred direction of spin. Rattlebacks are also known as "wobblestones" or "celts".

(MPG) or (CIN) or (AVI) - A home-made rattleback

(MPG) or (AVI) - Another pair of home-made rattlebacks

This shows how pretty-much any household object can be turned into a rattleback!

## Falling Chain

There's some really nice mechanics involved with falling chains. It is possible to make a chain fall to the ground faster than g=9.81m/s2 .

Don't believe me? Take a look at Andy Ruina's stuff (MPG) or (MOV) at Cornell. He does neat things with bikes as well

Andy also drew my attention to this amazing video "Fishing under ice" from Finland. Who would believe you can pour air out of a wheelbarrow?

## Kinematics

This program is useful for demonstrating the importance of visualizing and modelling the essential mechanics of motion - by example of an exercise machine.

(EXE) - An exercise machine

### Instantaneous centres - centrodes

The concept of instantaneous centres is illustrated here, with the moving centrode rolling without slip on the fixed centrode.
This is a simple mechanism: The blue bars are the mechanism, with a crank (rotating about a fixed point) connected to a follower with its other end moving as a slider along a straight line. The beautiful thing is that the motion of the follower can be reproduced exactly by the two rather strange shaped 'centrodes' rolling on each other. This is the subject of "kinematic geometry".

Do it yourself with these images for printing: matlab code

## Gömböc

Gabor Domokos and his Gömböc - as featured on the BBC "quiz" show QI, Fri 13 February 2009. You need to go about half way through (13m30s). Best viewed in RealPlayer, but it has also been known to work in Windows Media Player. (WMV)

If you make use of these, please acknowledge their source. I acknowledge the help and support of the following people: Prof Mark Warner (Cavendish Labs Cambridge) , Dr Paul Robertson and Dr Sue Jackson (Multimedia Group, CUED) , David Miller and the team (Frank, Gareth, Chris, Len, Gary) (CUED Mechanics Lab) , Prof Gareth McKinley, Prof Doug Hart, Pierce Hayward, Sean Buhrmester (MIT Mech Eng) , Emma Wilson (CUED)

This page contains low-resolution animated GIFs which should play automatically. To view the video full-size then click on the MPG , CIN or AVI links, choosing the one that plays best on your computer. If you're having trouble viewing the full-size videos then please go to testpage .