Page 2 Little g and big G and things MECHANICAL

After reading page 1 you can appreciate how important it is to do experiments in Physics - so let's start.

Big G, the gravitational constant that Newton introduced is a fundemental constant of the universe but little g is closer to home; it is the acceleration that any object will have if it is dropped off a cliff, say. So g is defined as the acceleration due to gravity. We all know that Newton's second law of motion tells us that -- Force = mass * acceleration. So any object has a force pulling it to earth with magnitude m*g .

It doesn't take much mathematics to see that, since Newton's gravitational
law gives F = G (M_{sun} m_{planet}) /
d^{2} then , little g will have a value

g = G* mass of earth/(radius of
earth)^{2 } since the earth's attractive force comes from it's
centre.

using values G = 6.67*10^{-11
}Nm^{2}kg^{-2}, ^{ }mass of earth
= 5.98*10^{24 }kg and radius of earth is 6.37*10^{6}
m we get g = 9.83 ms^{-2} .

Now, we can do an experiment to measure g -- a object (tin of beans if you like), a piece of string and a mobile phone which has a stopwatch as one of the applications ("apps") .

Secure the object at one end of the string and fix the other end to a support
that will take the weight of the object ( say the edge of a work surface
in the kitchen). On moving the object from it's rest position one can observe
an oscillating motion ie we have made a pendulum, and the oscillations will
last for a few minutes. Now all we have to do is find the time of one oscillation
T, and measure the length of the string from the centre of the object
to the support. An equation which is given in all Physics textbooks reads
as follows
--- ------ T^{2} =
4* ( pi )^{2}*L/g ----
so by knowing L and T the value of g is determined.

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My experiment gave the following results Time of ten oscillations
was 18.3 sec (remember !!!!start counting from zero) giving T = 1.83
sec and the length of the string was 0.80 m.
A calculated value of g is 9.42 ms^{-2}
.

A serious physicist will now ask "how reliable is the measured result?" The
timing of a single oscillation would be difficult to measure so the strategy
of timing 10 oscillations has given an improved accuracy. But one will see
that this time can have a value anywhere between 18.1 secs to 18.5secs so
there is almost a 2% inaccuracy with regard to timing. The measurement of
length is even more problematic as the one has to measure the distance between
the support and the centre of gravity of the object . The inaccuracy
here will be more like 3% so that a total accuracy of no better the 5% is
expected. Thus, the best we can claim is that g has a value between
9.0 ms^{-2} and
9.9 ms^{-2}.

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If you have chosen to complete the experiment then you have joined the company of illustrious scientists such as Galileo. You have also confirmed the notion that all bodies are acted on by the gravitational force. Aristotle had given the false hypothesis that matter on earth is made up of four elements - "earth", "air", "fire" and "water "- and this affected the manner in which they fell to earth. Stones would fall quickly as they had a lot of "earth" in them whereas a feather would fall slowly as it had "air" in it. The lunar Appollo crew (U-tube) made a convincing experiment on the moon in which a hammer and a feather were dropped from a height of about 1 m. As expected, both arrived at the surface of the moon at the same time and the only reason why the hammer and feather experiment cannot be repeated on earth is that air friction retards the motion of the feather so that a hammer/stone will always arrives at the earth's surface before the feather.

In January of this year (2014) an interesting article was published in the
Physics Education journal describing an inductive gravimeter (page 41, volume
49). By using electronic timing the free fall of an object could be investigated.
Full details are given in the article but the essence of the method is that
**the object is a magnet **and, as it falls through a series of coils
wound on a tube, an induced voltage is recorded on an oscilloscope . The
advantage of this experiment ( as opposed to the pedulum experiment above
) is that it clearly shows that the object is accelerating and therefore
g becomes very much part of the equations of motion.

A metre long cardboard tube was used, 45 mm OD and 40 mm ID. Fifteen coils, 10 turns each, were wound on the outside and were all connected in series; the spacing was 5 cm between the centre line of the coils and an illustration of two of the coils is given.

The illustration shows that the coils are series connected with the left hand wire continuing from coil to coil. The return wire on the right hand side comes from the fifteenth coil at the lower end of the tube (note the wire is insulated copper and has a diameter of 0.052 mm).

A Picoscope ADC100 is used to record the induced voltage and a "single shot" mode is used. Results are presented below and it is clear that these times could never be made with a stop watch.

The trace clearly shows that the magnet is moving faster and faster as it is acted on by the gravitational force.

The magnet N_S poles give rise to positive and negative swings in the induced
voltage and, from an EXCEL trace the times of the middle point can be determined.
We know that these cossings will occur just when the magnet passes the coil
so a time versus "fallen distance" of 0.05 m, 0.10 m, 0.15 m .......
can be plotted. From the laws of motion we have s =
0.5*g*t^{2} + u*t so the data can be fitted to a polynomial
of second order.

We therefore find that the falling object method gives
g = 8.4
ms^{-2} . One can only state
that no allowance has been made for air resistance in our theoretical framework
and, as we see with the feather, this can be considerable. Thus a value of
g = 8.4
ms^{-2} is an acceptable result.

Little g is the topic of a recent review article by James E Faller at the National Institute of Standards and Technology (NIST) which covers the subject in greater detail.

The measurement of Big G is sometimes neglected in student's Physics courses as this experiment requires quite a lot of patience and care. There was a flurry of excitment in the scientific community several years ago when it was rumoured that the gravitational force might not be exactly inversly related to the mass separation squared. Several elegant measurements were carried out in many Laboratories but all failed to show any discrepancy with Newton's formulation.

Leybold didactic have always provided apparatus of outstanding quality and their apparatus is usually used to measure G. Comprehensive leaflets are provided as illustrated:-

Newton's pronouncement in Principia of his gravitational theory was the staring point of science and we shall see that theory and experiment go hand in hand through the scientific story. With the discovery, recently, of the Higgs boson we have to realise that sometimes the experiments will get very complex and theory has to wait some years for verification. We all know that Newton didn't have the last word on gravity and that Einstein developed the General Relativity Theory which is currently accepted as a more complete theory for gravity. At a summer school in 2007 Martin Hendry presented a paper "An Introduction to General Relativity, Gravitational Waves and Detection Principles" which is a good 96 page read???? On page 88/ 89 he shows how the precession of planetary orbits follows naturally from Einstein's theory but was impossible to predict from Newton's Gravitational Theory.

Mechanics

Once the idea of forces and gravity was understood the wonderful subject of Mechanics could begin. A very helpful text book is "Mechanics" by R C Smith and P Smith Wiley 1973. Just two examples will be given here --- water power by an Archimedes Screw and Rocket propulsion.

Water power.Energy and Work amount to the same thing and we all know the work comes from moving a force over a distance. Energy from water is therefore generated by a mass of water, m, falling through a height h. As above, we wrote -- force = mass * acceleration --so the potential energy is given by m * g * h . Power is simply the energy generated per second as the following diagram shows:-

In one second the front end of the cube of water covers a distance 1m downstream
from the weir and the trailing end just manages to fall over the weir. So,
all the volume of water (inside the cube) falls 1 m and the mass can be
calculated knowing that the density of water is 1000 kg m^{-3} .
Also knowing that little g is about 10 we can see that the energy generated
in 1 sec is 10,000 Joules which, of course, is 10 kW. In each particular
installation of an Archimedes Screw the volume of water will be different
-- so we can write a general equation speed (m/s)* fall (m)*
width of channel (m) and then this volume is multiplied by the density (=1000
kg/cubic meter) and little g (=10). The purpose of the screw is to let the
water fall in an orderly manner and convert the potential energy into kinetic
energy of rotation and so "power" an electrical generator. The conversion
is moderately good at 20 % to 30 %.

A 40 kW installation is pictured below:-

Rocket propulsion

Do you have to be a rocket scientist to understand rockets? Possibly not, although I can well remember that most of our class found it difficult at school. Our main dilema was that we felt " there was nothing to pull /push against in space" . For instance someone in a swimming pool has the water to pull against to move forward and aeroplanes are pulling against the air even though this is not a very dense material. In space, however, there is nothing to pull against!!!!! Some long time after this school experience I watched the Open University videos when they were at an ice rink to demonstrate the law of conservation of momentum. It was then obvious that one "pulls/ pushes against the matter that is ejected". If there are a couple of people stationary on the ice and either person pushes the other person away then the video clearly shows that one person moves forward and the other moves backwards; momentum is conserved : it was zero before the push and zero after and that is how momentum has to be in space. The diagram below shows how 9 'bits of rocket fuel" are pushed away to give the green "capsule" plus astronaut a forward motion.

We see that with the first blast of fuel the rocket has a speed of v/9 .
Our coordinate system has been the same as that for any person watching
from the launch pad; both would observe a "before" speed of zero and
an "after" speed of v/9. Physics dictates that the momentum equation
is - "momentum before "= "momentum after", so we have 0 ===
mv + 9*m * V_{rocket} . Just like the situation
at the ice rink, the rocket moves with v/9 in the opposite direction to the
fuel.

Now, we link our coordinate system to the astronaut . He/ she will appreciate
that an impulse to a new speed of v/9 has occured (and things may rattle
a little on board the craft) but we can change our coordinate system to be
"normal" and take the speed as zero within
the spacecraft and the remaining 8 portions of fuel,
The first diagram can therefore be repeated and
a momentum equation is written down as before having realised that mass of
the astronaut plus fuel is now 9*m. ----- so----
0 = mv + 8* m*V_{rocket} and the rocket will move with v/8
in the opposite direction to the fuel. Now someone watching from the launch
pad will see the rocket going faster as it already had a speed of v/9 so
now it will have a speed of v/9 + v/8 . If this procedure is repeated, ie
updating our cordinates to be with the astonaut after each impulse then
all NINE fuel ejections can be covered and we find that the final rocket
speed is a summation: v/9+v/8+v/7+v/6+v/5+v/4+v/3+v/2 +1*v (=2.826
* v ) as viewed from the launch pad , The correct theory gives a formula
(page 166 Smith and Smith ) V_{rocket} = v * Natural Log (
1 - 9/10) == 2.3026*v .

Note, if the fuel had been ejected all at once the rocket speed would have
immediately jumped to 9*v but neither rocket nor astonaut would have survived.
If two fuel burns had been employed, 4.5*m ejection at first would
give 0.82*v for the rocket. Then the next 4,5*m would be separated
and this would give 4.5*v in "capsule " coordinates so we see that the final
rocket speed , V_{rocket} , would be 5.32*v. It is therefore not
unexpected that a continuous burning of fuel will lead to 2.3026*v as given
by Smith and Smith. They also provide the full theory of rocket propulsion
on pages 166 /169 and show that it is necessary to have a two stage rocket
to leave the earth's gravitational field.

Mr Richard Branson is, no doubt, well versed in rocket science and if the Virgin Galactic space vehicle makes it's maiden flight this year (2014) then it will be the start of an era "space flights for all" and we will all have to become well versed in rocket science (see guardian weekend 22-02-2014).