Extra Info
What is a Day
The energy needed to move the Earth 1 degree of rotation is very close to the energy used to move the Earth for 1/360 of the mean solar day, which is 240 seconds.
The mean solar day is based on the time it takes to see the Sun in the same arbitrarily chosen position in the sky, for example at mid-day on the following day. It is not exactly 24 hours each day, there are seasonal and longer term variations, partly due to the orientation of the Earth and its elliptical orbit. It takes on average 24 hours.
The star day or sidereal day, the time to rotate 360 degrees, measured by using other star systems, external to the Sun, as reference points and it is 23 hours, 56 minutes, 4.1 seconds. So 1 degree of rotation takes 239.33 seconds.
There are also measurements known as solar days, stellar days and others. Each system has its own uses, but the sidereal day is seen as being more exact for describing the mechanical motion of the Earth as a planet on its own, away from the Sun. The angle of the notional axis rotation of the Earth also wanders and wobbles and precesses adding ever more delightful complications, each of which reveal more detail of the workings of our planet. The spin of the Earth is also slowing down.
To measure that one degree of rotation, if all we need is an overall comprehension, 240 seconds or 4 minutes is easier to remember, faster, simpler and clearer than 239.33 seconds. If instead, we are having to guide something vast or heavy to fit carefully into or between something else and not have catastrophic damage, then we do need the fine grained, nit-picking details, they are vital. Awareness is everything.
Be aware that the solar day itself is an average and varies, it is used as a notional amount. Be aware that even a small difference when multiplied by something big can itself get very big and remember that as a possible factor for future use. We are about to go big in many of our calculations. Simpler statistics are more useful initially. Later, if something at first appears to be wrong, we should return and look to the fine grained details.
What is a Radian
Radians, Units and Joules
The units that a Joule comprises are;- kg, m2, sec-2. There is no "radian squared" to see, why? It is reasonable to ask why should the units not be kg, m2, radians2, sec-2 ? A radian has no units because it is a notional fraction, a number applied in reference to a circle. It is a ratio directly related to the ratio of the radius, r, to the circumference, c, of a circle. The formula goes c = 2π r, where π is a special number. There are 2π radii, or arcs of length equivalent to 1 radius in the circumference of a circle. This, admittedly, is a bit "circular". Let us progress by degrees.
If a circle were to be sliced like some fantastical cake into 360 equal pieces, they would each span 1 degree of angular measurement. A degree of this kind is a 1/360th of the "going round"-ness of a circle. The angle subtended by an arc of length 1 radius along the circumference of a circle is 360/2π degrees, or 57.296 degrees and is referred to as a radian. It is a useful amount for doing many calculations that involve, bending, turning and vibrating.
Both π and radian are transcendental numbers, the more you look and the finer detail you can focus into, the more they keep going. They don't seem to stop, when you try to limit or define them by numbers. We shorten π to 3.14, or sometimes (22/7), but it is a matter of pride to some people to be able to recite π to many places of decimal precision. Depending on the use required, a suitable precision can be readily chosen and calculated.
There are several methods for calculating π, and so giving a numerical value to a radian. Sir Isaac Newton developed what is probably the most elegant method.
Celestial Mechanics : Rotation
To get an idea of the scale of the energy embodied in the planet and then to be able to compare it with other amounts of energy, let us use some statistics readily available on the internet. Depending on the sources used, slightly different figures can be found, but the overall flavour of the energy amounts is still on the same huge scale.
The kinetic energy or "energy-of-movement" of the Earth is made up of all the energies that affect it, both internal and external. Two expressions of those energies are how the planet spins and how it orbits the Sun. To get a figure for the energy of rotation, we need to know the mass, the size, shape and spin speed of the Earth. These numbers are next combined in two formulae to get a value for the kinetic energy.
The mass of the Earth is popularly given as M = 5.972 × 1024 kg..
The equatorial radius is Requ = 6,378,137m = 6.38 × 106 m.
The mean radius is Rmean = 6,371,000m = 6.37 × 106 m.
The first formula is for the moment of inertia, which relates what force is needed to overcome an object's happiness just to stay as it is and where it is.
The moment of inertia, I, of the Earth is given as both 2/5 MR2 , arrived at by devious calculus for a solid sphere shape, or as 0.3307 MR2 arrived at by also considering rigidity of the Earth and how it spins in relation to the gravity effects of the Sun, Moon and planets relative to the equatorial radius of the Earth. These two figures do ignore other details such as the variation of density with depth and other aspects of the polar flattening of the shape of the Earth. So we get a range of values from which to choose a favourite.
I simple = 9.70 × 1037 kg m2. I notsosimple =8.03 × 1037 kg m2.
The next formula, that for the kinetic energy combines the moment of inertia, I, with the speed of how fast the Earth is spinning, its angular velocity.
The rate of rotation is 360 degrees in 23.93 hours. We convert this to the angular rate in radians per second.
The angular velocity, ω ,of the Earth is given as 7.29 × 10-5 radians per second.
The kinetic energy of a rotating sphere is given as I × ω2/2 .
Combining the values above, the kinetic energy of the Rotation of the Earth is a "simple" 2.58 × 1029 J or "not so simple" 2.14 × 1029 J.
Celestial Mechanics : Orbit
To get a figure for the orbital energy, we need to know the mass, the size, shape of orbit and orbital speed of the Earth.
The mass of the Earth is often given as M = 5.972 × 1024 kg..
The mean orbital radius is Rmean = 6,371,000m = 6.37 × 106 m.
The moment of inertia, I, of the Earth is given as MR2
The rate of orbit is 360 degrees in 23.93 hours. We convert this to the angular rate in radians per second.
The angular velocity, ω ,of the Earth is given as 7.29 × 10-5 radians per second.
The kinetic energy of a rotating sphere is given as I × ω2/2 .
Combining these values, the kinetic energy of the Rotation of the Earth is 2.14 × 1029 J.
Chain Chatter
A cranking gear, also called a sprocket or sprocket wheel, for a chain driven system can have many sizes and many teeth. The more teeth a gear can reasonably have, the smoother the chain will move. Chain chatter happens with imprecise machine parts and when the tooth number is low so that the gear has too few teeth.
Practical considerations for a chain and its gears and sprockets are the masses, weights and forces involved and the physical details of materials proposed for the machinery. Bicycle chains and gears have and still are evolving, mainly around steel alloys, so that the overall range of specifications is well defined. A bicycle chain will not usually be 10 metres long, made of rubber plates, with 3 cm diameter aluminium roller bearings, glass rivets and have to support moving a 1 ton weight. Different desires will drive the evolution of the chain. Light-weight yet strongly tensile steel with anti-corrosion ability, perhaps hardened Boron steel, spacing between rollers, say about 13mm, roller width 3 to 5 mm diameter, about 4mm space between the roller plates etc., such are actualities of typical bike chains. The gears associated with a drive chain also co-evolve, as do the structure and materials of the rest of the system, which here is the bike itself. Idealized and exotic materials can be found and used, but in the main, economical and practical solutions have to be found easily and readily for most bikes and other machinery. Common constraints often apply. A bamboo bike can have aluminium sprockets with a titanium chain and weigh in at about 10 kg which allows it to carry a 100kg passenger over reasonably rough ground downhill at perhaps 50 kmph. A 60,000kg battle tank whizzing at 70 kmph has larger sprockets and chains, but many common constraints. Chain chatter, which is what we are singling out here for examination, is definitely one of them. The sticking-out bit on a sprocket wheel is where the chain's roller bearing first becomes engaged to the sprocket and later, after being driven, leaves the sprocket wheel.
We examine here, in a simplified way, one aspect that contributes to chain chatter.
Consider that the gear wheel can be characterised as a regular polygon, of 3,4,5,6,7,...up to n sides.
The biggest circle that you can fit or inscribe inside the polygon has a radius smaller than the radius of the polygon from its centre to one of its corners. The corners of the polygon stick out proud from the circle.
We use Pythagoras theorem and trigonometry to work out the difference between the two radii.
If we draw a radius to the point where one of the sides of the polygon touches the circle, that will be at right angles or 90 degrees of arc to the side of the polygon all of which are tangents to the circle. If we take a radius of the polygon on either side of this radius of the circle, we get 2 sides of an isosceles triangle, which is made up of two right angle triangles inside it. The shortest side is half the length of the side of the polygon. The next longer side is the radius of the circle and the longest side is the triangle's hypotenuse, which is also the radius of the polygon.
The angle subtended at the centre of the polygon by one of its sides is given by 𝜃 = (360degrees/ n) x (1/2), = 2𝜋 radians/(n x 2), = 𝜋 /n.
The length of the hypotenuse is given by h = r x cos(𝜋 /n), so the difference between the two radii is r - r cos(𝜋 /n), = r(1- cos(𝜋 /n)). Looking at how cos(𝜋 /n) changes as n increases from 3 to something big, we see that (𝜋 /n) ranges from 60 to 0 degrees and cos(𝜋 /n) ranges from 1/2 towards 1 so that (1- cos(𝜋 /n)) ranges from 1/2 towards 0. The more sides the polygon has, the smaller the difference between the lengths of the two radii and also the smaller is the effect of this difference towards chain chatter. Balancing against this is that the diameter of the chain bearings has to be decreased as the number of sprocket teeth is increased in order to physically accommodate the teeth on the sprocket wheel. The chain bearings can only be reduced until they do not go beyond the specifications needed to service the load on the system.
