B8. Motion on precribed trajectory
Newton's law of gravitation. General attraction of masses. Gravitational constant
Newton compared the Moon's attraction by Earth with that of a body on Earth's surface by Earth, that is, with its weight and drew the conclusion: The weight of masses on Earth has the same cause as the maintenance of the Moon in its orbit about Earth and that of the planets about the Sun. However, this attraction is mutual. All bodies on Earth have weight - its quantity depends only on how much of it is there, not on their specific properties such as their chemical properties. - Weight, that is, the property of being attracted by Earth, is therefore a property of matter. But this attraction is mutual. Therefore matter must possess the two properties of attracting Earth and of being attracted by it to the same extent. Since Earth consists of the same matter as the other bodies, weight is a confirmation of mutual attraction of matter. Weight on Earth, attraction between the Sun and the planets, attraction between Earth and the Moon - all of them are confirmations of one and the same force.
Hence Newton drew his conclusion regarding mutual attraction of masses, the gravitation of matter throughout the universe: Attraction is an inherent property of matter, that is, any two mass points attract each other with a force f which is directly proportional to the mutually attracting masses m and m' and inversely proportional to the square of their mutual distance r, that is, f = (m·m')/r2)·K (Newton's law of gravitation 1683). If each of two masses possess unit mass, that is, m=m'=1 g, and they are 1 unit length apart, that is, r = 1 cm, then f=K dyn. The force of attraction of two masses can be measured and turns out to be 6.68·10-8 dyn. The number K is called Gravitational Constant. The equation for f shows that its dimension is cm3g-1sec-2.
Using this value of K, you discover that, for example, two masses of 1 kg each at a distance of 10 cm from each other attract with a force which is equal to the weight of 0.00068 mg*.
While studying the
general nature of physical forces, especially of those forms
which arise during actions
at a distance, Faraday said in 1858:"In Newton's words: "Weight of matter
is inherent, permanent and essential, so that a body can act on
another one through vacuum over a distance without any mediation
by something else, as a result of which and through which its
action and force is transferred from one to the other".
This is an immense absurdity. Weight must be caused persistently through an agent according to certain laws; however, whether this agent is material or immaterial, Newton leaves to the discretion of the reader , , , , Those people who acknowledge Newton's law, but not follow him further, understand by attraction that matter attracts matter and that this attraction is inversely proportional to the square of the distance.
Imagine now a material mass (or a molecule), assume that for this purpose the Sun is suitable and imagine that a sphere of the size of a planet, say our Earth, has arisen suddenly or been brought from a great distance to its actual location in relation to the Sun, when Gravitation will act and we will say that the Sun attracts Earth and simultaneously Earth attracts the Sun.
However, if the Sun attracts Earth, this force of attraction must either be a consequence of the presence of Earth in the vicinity of the Sun or it must have existed previously in the Sun before Earth arrived. . . Consequently, the force around the Sun and throughout infinite space must have existed all the time, irrespectively of whether other bodies, on which gravitation can be exercised, exist or not; however, this force must exist not only around the Sun, but around every existing material particle. This concept that the capacity to act in space is persistent and necessary, also with respect to the Sun when Earth is not in its place, and that a certain gravitational action, when Earth is in its place, is the result of that state, I can conceive to be in harmony with conservation of force; I suppose that this was Newton's concept of gravitation which philosophically is just the same as the generally adopted idea regarding light, heat and radiation and which (in a yet more general and comprehensive sense) now has been drawn to our attention in a specially forceful and deductive manner by the phenomena of electricity and magnetism as a result of their dependence on mutual forms of force" (Exper. Res. Vol. 3 p.525 of the German edition).
You speak of fields of force whenever the force under consideration varies continuously from place to place and is given at each location by the value of a function. The centrifugal forces inside and on the surface of a rotating body have such a field type distribution through the entire volume of the body; there is no reason why the field should not be extended beyond the surface of the body, for example, beyond Earth's surface into the atmosphere as the centrifugal field of Earth. .
Testing of gravitational law. Density of Earth. Weight of a body below and on the surface of Earth.
Newton derived his law only
from cosmic phenomena and only tested such phenomena
(and explained
Kepler's laws and the tides as being consequences). Cavendish was
in 1798 the first to also prove mutual attraction of masses by
laboratory experiments and to measure it (Fig. 101): He employed two small metal balls
(each of 730 g) at the ends of a horizontal, wooden bar
supportedby a threa , to which he could move two large spheres of
lead ( of 158 kg each) by means of a rotatable frame. If the two
large spheres (seen from above) are at AA, the small
ones remain at rest, being attracted by equal forces in
opposite directions. However, if you place the large spheres, for
example, at BB, the small spheres move towards them. He
measured the deflection of the wooden bar. -
From this experiment, you can also compute Earth's mass. If the one sphere has the mass m, the other the mass m' and a is the distance between their centres, which can be considered to be the actual points of attack , then m attracts m' with the force f = m·m'/a2·K. Earth of (unknown) mass M and radius R exerts on m' the force (M·m'/R2)·K. However, Earth's attraction of m' is its weight, that is, m'g =.(M·m' /R2)·K, whence f/m'g = mR2/Ma2. All quantities are now known, but f and M. However, Cavendish's experiments had already determined f, whence Earth's mass could be found. Moreover, since Earth's volume is known, its density could be computed: It contains about 5 - 6 times the mass of an equal volume of water. - These experiments also yielded the gravitational constant K.
With an arrangement similar to Jolly's double beam balance, Richarz and Krigar-Menzel 1898 have determined Earth's density at 5.51; their value was still considered to be the best in 1935. They compared the weights of two nearly equal spheres, one of which, A, hangs on the one arm above, the other, B, on the other arm, below a mass of 100 000 kg of lead (Fig. 102) . The attraction of the mass of lead increases the weight of A by the same amount by which it decreases the weight of B; the deflection of the balance therefore corresponded to twice the amount of the attraction.
By Newton's hypothesis, the weight of a body on Earth is the resultant of the attraction of each mass point on Earth of each mass point of the body. The magnitude of the force, by which Earth attracts a body, that is, its weight, must therefore be proportional to its mass, as is indeed the case.
According to Newton's hypothesis, the weight of a body must drop - by a quantity which can be computed - if the body is raised over Earth's surface, because thereby the mutual distance becomes larger. Also that is true: The same body weighs less high above Earth's surface than it weighs close to the surface. A balance was installed in a tower 25 m above its base on each scale of which hung 21 m below it, that is, near its base, yet another balance scale. The scale showed 0.01 mg with a load of 5 kg. The same 5 kg of mercury weighed at the height of the tower 31.685 mg* less than at the base; the theory demanded 33.000 mg* (Jolly).
How does the weight of a body change after it is taken below Earth's surface, for example, at the floor of a shaft? For the answer, we will employ (without proof) two theorems discovered by Newton. Imagine a quantity of matter to have been formed into an extremely thin spherical shell, a spherical homogeneous skin of mass or a mass bladder; what is the magnitude of the force of attraction of the spherical shell on a mass point? Answer: If the point lies outside the shell, the attraction is exactly as large as if the entire mass of the shell were concentrated at its centre; if the point lies inside the shell, the attraction is zero.
You can imagine a homogeneous solid sphere and similarly a homogeneous thick spherical shell, as they really exist, to have been composed out of very thin concentric homogeneous spherical shells. Hence you arrive at the conclusion: A homogeneous solid sphere attracts an external point equally strongly as if its mass were concentrated at its centre; the attraction of a spherical shell on a mass point inside it is zero.
Assuming that Earth is a homogeneous sphere, you arrive at the result (Fig. 103) that the point M inside Earth is not attracted by the sphere MA and receives from the sphere with radius OM the same attraction as if the mass of this sphere were concentrated at Earth's centre O. The calculation shows that the attraction at M is proportional to the radius, that is, the distance from Earth's centre. However, this means that the weight of the mass is largest at Earth's surface, decreases towards its centre (proportional to its distance from it and vanishes at the centre. But Earth is not homogeneous. Its deeper layers are, in general, substantially denser than the layers in the neighbourhood of its surface. Indeed, g is larger at the base of deep shafts than at Earth's surface (Experiments with pendulum ).
On the sea shore and inside certain river mouths, the water sinks twice daily and rises again - these phenomena are called ebb and flood, both together ocean tides. The time of high water - from one to the next elapse 12 hours 24 minutes - is closely related to the passage of the Moon through the Meridian, and its height also to the phase of the Moon, that is, the simultaneous position of the Moon relative to Earth and the Sun.
Around the times of full and new Moons, when the Sun, Earth and the Moon lie in one line, the difference between the highest and lowest water level is largest (spring tide, for example, at Portsmouth, it is about 4.1 m), at the first and last quarters, when the Sun and Moon form with Earth a right angle, the smallest (neap tide, for example, in Portsmouth, 2.3 m). The theory of the tide generating force is due to Newton (1687); it is one of the strongest supports of gravitation theory and all later work on the tides has been based on it, because they explain the tides satisfactorily by the attraction, exercised by the Sun and Moon. For its description, assume for the time being that only Earth and the Moon are present - the generation of the solar tide explains itself once you understand the lunar tide - and return to the attraction of the Moon by Earth.
Earth and the Moon attract each other: As the Moon moves towards Earth , Earth moves towards the Moon. Earth is not at all a centre at rest relative to the Moon's orbit and, strictly speaking, the Moon does not move around Earth, but the Moon and Earth travel together - once every month - about their common centre of mass . Since Earth has 80 times the mass of the Moon. [this point lies very close to the centre of Earth, only 3/4 times Earth's radius from it, near enough to entitle us to neglect frequently the orbit of Earth about it and to talk about the orbit of the Moon about Earth like about a centre at rest; for the generation of the tides, the orbit of Earth about this point has the main role, because the centrifugal forces, arising from it, yield a component of the tide generating force].
Earth moves towards this point as the Moon comes from the opposite side, however, without coming closer to it. Earth's sections, facing the Moon (59 Earth radii away)are pulled more strongly than its centre (60 Earth radii away), its other side (61 Earth radii away) more weakly than its centre; the former move more quickly, these fall more slowly towards the Moon than the centre.
The rigid part of Earth reacts naturally as a whole to the attraction - but the oceans react differently. The water particles yield each on its own to the attraction. The Moon M in Fig. 104 attracts the water at V more strongly than at C, that is, it tends to pull it away from C, that is, to raise the sea level at V; at the same time, it attracts C more than the water at J, that is, to pull the centre of Earth away from J, that is, to deepen the sea under J and correspondingly elsewhere to reshape the sea. However, the attraction is not the only force, which acts on the water.
On very water particle acts still the centrifugal force, which is generated by the motion of Earth about the centre of revolution, common to Earth and the Moon. (We neglect the centrifugal forces due to Earth's rotation about its axis.) The resultant of these two forces (the Moon's attraction and the centrifugal force) shifts the water particles; it is the tide generating force. Fig. 104 displays the tide-generating force at different points of a Meridian; the circle is a cross-section of Earth at a Meridian, DD are the Poles; the Equator passes through V and V', the Moon is at a large distance in the direction of M, the arrows indicate by their direction and size the tide generating force.
However, only the horizontal component, that is, the component which is parallel to Earth's surface must be considered; only this component causes the water to flood; although the vertical component can make the water appear to be somewhat lighter at V and J and somewhat heavier at D, it is not large enough, in order to overcome gravity and to contribute to the motion of the water. That is why the component, vertical to Earth's surface, has been omitted. We obtain thus, as indicated perceptively in Fig. 105, the system of forces, distributed over Earth's surface, which tempt to drive the water horizontally towards the two end points V and J of the diameter of Earth, at the extension of which is the Moon. At V and J, the water rises above the surrounding ocean; on the Meridian through DD, which is perpendicular to the diameter VJ, the sea sinks simultaneously right around the entire globe below the mean sea level. (Note that at V and J, towards which the water is driven and above which the flood wave is highest, the horizontal component is zero as well as at the Poles D. Moreover, the side facing the Moon changes steadily as a result of Earth's rotation, that is, the pattern of Fig.104 moves therefore steadily about Earth).
You obtain in this way a clear picture of the origin, magnitude and direction of the tide generating force; however, for a prediction of the course of ebb and flood, for example, for a prediction of the time at which at a given location of the ocean high water occurs, not much has been gained: Earth's surface is not covered everywhere - as it has been assumed here - and also not covered to the same depth by the sea; beyond all, the ocean does not attain instantaneously the equilibrium state which corresponds to Fig. 104 and which it would assume, if the moved water masses had enough time to do so.
Nevertheless, one has
succeeded, on the basis of theory and observations, to compute
for given locations tide
tables, which
predict the
time of occurrence and the height of the tide very exactly. One of the most successful methods of
prediction is the harmonic
analysis of the tides (William Thomson 1872): Not only the Moon, but also the Sun contribute to the rise and fall of the
ocean - the height of the solar component is half that of the
Moon - as well as other periodically recurring phenomena have an
effect. The wave of the tide which passes a given location at a
given time is a superposition
of several partial waves. If you know the time of high water and the height of
one of these partial waves on anyone day at a given location, you
can predict the water level, as far as it is due to this wave,
with certainty for every instant at this location.
"There is no choice for a given composite wave as it can only be constructed in one manner. You have here a complete analogy with musical sounds; a musical sound of any kind consists of a fundamental tone and its overtones: octave, duodecimal, etc. In the same manner, one views the tide wave, which has become irregular, to consist of a fundamental flood and higher components of half, third, etc. wave lengths. The periods of these higher harmonics also are halves, thirds, etc. of that of the fundamental flood.
The prediction, based on the concept of harmonics of the tides must unavoidably fail, if we have not uncovered the real causes, for example, whether the overtone with a certain period exists only in our imagination, but does not correspond to physical reality. The success of the prediction therefore is a proof for the truth of the theory. If you take into account that the incessant change of the tidal forces, the complicated shape of our coasts, the depth of the ocean and the rotation of Earth around its axis are all part of the problem, we may consider the good prediction of the tides to be one of the greatest triumphs of the theory of general gravitation". (George Howard Darwin 1845-1912)
b) Moving rigid body, which can rotate about a fixed axis
The influence of the distribution of mass of a body around its axis
What happens in a body's cross-section, perpendicular to its axis of rotation, at one mass point, happens at each mass point (Fig. 93); it occurs at each cross-section irrespectively of the point of the axis at which the body is intersected. However, this is only true qualitatively; quantitatively it may differ greatly depending on the manner in which the mass of the rotating body is distributed about its axis of rotation.
This statement means the following: The expression F=mrw2 for the centrifugal force shows that mass points with equal mass m and located equally far from the axis of rotation generate equally large centrifugal forces (since w is the same for all of them) and apply equally large tensile forces on the axis. If two points lie diametrically with respect to each other, their centrifugal forces cancel each other and leave the axis unchanged. Is the entire mass distributed symmetrically about the axis so that on each line, perpendicular to the axis, points at both sides of the axis correspond in mass and distance, then the axis does not experience centrifugal effects.
We then say that the axis is in equilibrium. For example, this is so for a circular, homogeneous, flat-faced disk which rotates about a central axis, perpendicular to it. However, if the disk is not homogeneous, say, one half of the circle is out of wood, the other out of lead, then the centrifugal forces mrw2 of the leaden half are larger than those of the wooden half, since mass points of lead contain more mass (m) than those of wood, The centrifugal forces acting in the same diameter do not cancel each other, the axis is not in equilibrium, it shakes the bearings of the axis. The form and size of the disk are the same in both cases, but their mass distributions about the axis of rotation differ; their different behaviour with respect to the axis arises out of the mass distribution.
We have used the two disks as an example, because it displays clearly what is of importance here (the mutual compensation or lack of compensation of the centrifugal forces around the axis of rotation).We now proceed to a cross-section perpendicular to the axis of an arbitrary rotating body.
So far, we have only spoken of how two points can cancel each other in their centrifugal action on the axis of rotation. Obviously, the centrifugal forces which two points generate can together cancel their effect with a third one. Those two points and the third one must then have such distances from the axis and be located relative to each other so that the resultant of the centrifugal force of the first two cancels the centrifugal force of the third. It is a mathematical task to place through a body of given shape a line such that the centrifugal actions in all sections together - that is, in the entire body - cancel each other with respect to this line. In that case, this line is an axis of rotation in equilibrium.
A thorough examination of this problem shows that there exist in each body three such lines, that they pass through its centre of gravity and are perpendicular to each other. They are the principal axes of inertia of the centre of gravity. (stable axes of rotation )
If the axis of rotation is not such a principal axis, one part of the applied centrifugal forces remains uncompensated and therefore the body shakes in its bearings. It is a special case, when the axis of rotation is a principal axis. In general, a part of the centrifugal forces remains uncompensated. This uncompensated part tries to turn the axis about its support point. (Note: not only to turn, but also to redirect or even turn it over!).
Also this only intended rotatory motion can be represented by a rotor; you must imagine that it is placed along the axis of the intended rotation, that is, along a line which also passes through the other point of support. This vector forms therefore with that of the already present rotation an angle. The sum, formed according to the parallelogram rule of the two vectors yields the axis, about which the body would rotate, if it could execute both rotations simultaneously (that is, if the upper point would not hold it fixed). This axis of rotation therefore forms with the actual one an angle. We see thus, and this is the important point: If an axis of rotation is not a principal axis, the rotation of the body generates an impact perpendicular to the axis of rotation,which make the bearings of the axis ineffective, but to which also a rotor must be allotted.
This different nature, with which the rotation of one and the same body can occur, demonstrates the importance of the mode of mass distribution relative to the axis. If the body is displaced as a whole, then single mass points have no role, only its total mass - if you know its size, you can compute from it and its acceleration the force to move it. However, if the same body rotates about an axis, the mode of distribution about the axis or, in other words, the distribution of the distances from the axis of the individual mass points, is decisive. It is no longer enough to deal with the concept of mass, you must introduce the new concept of moment of inertia.
The rotation of a body discussed earlier was at uniform angular velocity. However, if the body rotates as a result of lasting action of a force, its angular velocity is accelerated. Just as we have determined out of the magnitude of acceleration that of force - we had to introduce for this purpose the concept of mass! - we pose now the question regarding the relationship between acceleration of angular velocity and the magnitude of the rotating force.
The nature of rotation creates here a difficulty. If the rigid body is free to move, all its points have the same velocity, if it can rotate about an axis, the velocity of each of its points depends on its distance from the axis. No longer can you imagine the entire mass of the body to be concentrated in an arbitrary point, but you must take simultaneously into consideration all its points. Moreover, as long as one is concerned with a uniquely determined axis, each point has a unique distance from the axis.
However, an axis
is a line, which is determined by any two fixed points of the
body. If we fix one time these, another time those two points of
the body, the axis is every time another one and every mass point
(the point a in Fig. 106) has each time another distance from the axis. In other words: In
a rotating body, you must take into consideration the position of
every individual mass point relative to the actual axis of
rotation. For this
reason, we cannot only use
the concept of mass and must introduce a new quantity which has
for rotation the same role as has mass for translation. This
concept is the moment
of inertia (the term is due to Euler).
It is a quantity which can be measured physically and is characteristic for a body which can rotate about an axis. However, since a body can be made to rotate about an infinity of axes, it has infinitely many moments of inertia. Its determination - as that of mass by weighing - is in each case a matter of an experimental procedure (not to be discussed here). For (homogeneous) bodies, the form of which can be described mathematically, such as a cylinder , sphere, ellipsoid, etc. , it can be computed.
We obtain the formula for the moment of inertia as follows (Fig. 107): LM is a mathematical straight line. which can rotate about a vertical axis, that is, in the horizontal plane; m1. m2, m3 are mass points on it at distances which cannot be changed and at the distances r1. r2, r3 from the axis. From a state of rest, LM has reached in some manner or other, at the end of the first second in uniformly accelerated motion, the angular velocity w and is then left to itself. The three points have then the path velocities r1w, r2w, r3w, that is, if they were at this instant freely movable, they would move on at these velocities tangentially to the described circles. Since they started from zero speed and have reached this velocity at the end of the first second, r1w . r2w ,r3w are their path accelerations, whence m1r1w . m2r2w , m3r3w are the forces, which have acted on them during this second.
The forces m1r1w , etc., are rotating ones. Every single one can be replaced at corresponding distances from the axis by another force of arbitrary magnitude. However, you can replace - we assume this to have been proved - also more than two simultaneously and in the same direction rotating forces by a single one, where the rotating moment of the single one equals the sum of the rotating moments of the several forces. The sum of the rotating moments, with which we are concerned here, is: (m1r1w r1 + m2 r2w r2 + m3 r3w r3) or
The index n refers to every number in the sequence 1, 2, 3, 4, etc. from 1 to the number which indicates how many mass points m ( and distances r from the axis) are involved; in the present example, it runs from n = 1 to n = 3. The symbol S indicates that the terms mnrn2 are to be summed.
In order to replace this sum of rotating moments by the moment of a single force R, you must apply R at such a distant a from the axis that
If you now still demand that the force should be applied at the distance a=1 cm from the axis, it must have a corresponding magnitude at this distance, say R1. We reach thus the result: In order that the rigid straight line with the three mass points m1. m2, m3 shall attain at the end of the time unit the angular velocity w, while it rotates about the axis AA, there must apply at the distance of 1 cm a force R1 of the magnitude
This is the relation between the rotating force and the angular acceleration w - but only with respect to the axis AA.
