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B7. Motion on prescribed trajectory
Demonstration of centrifugal force
Between the body which the centrifugal force does not change and that, which it destroys, is the body which it only deforms. A good example - at least in its essential aspects - is the centrifugal regulator of Watt 1784, who invented it for the regulation of the supply of steam to engines (Fig. 95). The vertical axis W carries easily moved, jointed arms which end in heavy spheres; if the axis rotates with an appropriate angular velocity, the spheres move apart centrifugally. They pull a sleeve H on the axle upwards and activate thus the steam valve, which is linked to the sleeve by a rod G.
Spreading by centrifugal forces becomes especially illustrative in the case of fluid.which is removed from gravitational effects and is in a state of total rest and equilibrium, and forms a sphere. If you rotate it about one of its axes, for example, a sphere of olive oil (Plateau's experiment), floating in a mixture of alcohol and water, the sphere becomes an ellipsoid. The centrifugal forces due to the rotation compete with the very slight cohesive forces, which hold the fluid together, and cause deformation by driving the individual mass points outwards away from the axis.
Of the points which lie on meridians - each of which converts during the rotation from a circle into an ellipse - those on the equator are driven furthest away, those at the poles do not leave the axis, those in between the equator and the poles move correspondingly far outwards, depending on their distance from the axis of rotation.( Obviously the centrifugal forces are the larger the further away they are from the axis). The sphere is stretched outwards perpendicularly to its axis. As a result, the poles move closer together, the sphere flattens at the poles and becomes an ellipsoid of rotation (Earth, Moon, the planets are spheroids, flattened at the poles).
A similar explanation concerns the deformation of the surface of a fluid in a vessel which rotates about its vertical axis (Fig. 96). The initial, horizontal, free surface of the fluid becomes a paraboloid (with the axis of rotation an axis of symmetry). The force which attacks each of its points is the resultant of the centrifugal and gravitational forces. The surface must take on a shape in which at each point it is perpendicular to the forces acting there, since only then will the surface maintain itself. If the forces had a perpendicular component (tangential), the fluid would follow it until the surface is perpendicular to the acting forces.
Magnitude of centripetral and centrifugal forces
In order to compute the magnitude of the centripetral force, you start from the fact that it pulls the mass point away from that line which it would describe if it could follow its inertia. If this pull were to vanish as soon as the mass point arrives at D (Fig. 97), the point would continue to move from D along the tangent with the velocity which it has at D. But the pull continues to act, deflects the point away from the tangent and forces it to continue along the circle. You can compute the magnitude of the centripetal force from the magnitude of the deflection during the time t.
Let the angular velocity w be uniform. the mass point be at the distance r from the axis of rotation; its path velocity is then v=r·w, the velocity at which it tends to proceed from D tangentially and uniformly. If it moves during time t from D to E, then DE = t·v (because it travels in unit time the distance v). Now make t so small that the arc DE = t·v. Its deflection from the tangent during t is GE or also, drawing to it a parallel through E, DF (= GE).
The force F which causes this deflection acts continuously on the point, that is, accelerates it. Let
the point have the mass m, so that it receives the acceleration F/m, that is, it moves during t the
distance (1/2)(F/M)·t^{2}, whence the distance DF = GE. By a known theorem of
planimetry:
= 2r·DF, that is, v^{2}t^{2} = 2r·(1/2)(F/M)·t^{2},
whence F = mv^{2}/r. And since v = rw,
then F=mrw^{2}= 4(p^{2}/T^{2)}mr
[ introducing the time T of a
complete circuit and remembering that a point at the distance r from the axis travels in time T the distance 2pr,
that is, in unit time the distance 2p r/T, but
the distance covered in unit time is also rw, whence
rw = 2pr/T].
Hence the magnitude of the centripetral force,
which maintains the point on the circle, depends on the mass, the radius of the circle, that is, the distance from the axis, and
the path velocity of the point which moreover depends on the angular velocity w.
If the rope in Fig 94 has the length of the seconds pendulum and the body travels around the hand 10 times per second, then the rope is subject to 400 times that load which it carries when the body simply hangs. In fact: The centrifugal force is F = 4p^{2}rm/T^{2}; if l is the length of the seconds pendulum, you set r = l. If the weight of the body is P, then m = Plg, and, since g = p^{2}l (according to the equation of the pendulum), then m = P/p^{2}l . If moreover T = 1/10 sec, then finally F = 400P.
The centrifugal field, generated under these conditions is therefore 400 times as strong as the gravitational field. People have succeeded in generating such fields in excess of 100 000 times the gravitational field (Ultra-centrifuge) When the circumferential velocity is large enough, the centrifugal force will always overcome cohesive forces of bodies, that is, it will destroy them.
Planetary motion - General attraction of mass
Central motion (Motion of planets), Copernicus and Kepler
As long as the mass point, which travels around the point C of an axis (Fig. 93), is rigidly linked to it, the point cannot follow any other path but a circle. However, if the connection is interrupted while it circulates, it flies off along the tangent like a stone swung in a circle as the rope tears (Fig. 94). If the stone , after the rope is broken, were to continue on its orbit, we would look for an explanation in the form of a force, which takes it away from the tangent into a motion around the hand. If it has the mass m and describes a circle with radius r at uniform velocity during the time T, we would say that the stone moves as if the hand exerts on it a centripetal force of magnitude 4p^{2}rm/T^{2}.
But we could say this only provided the stone follows a circle and at uniform velocity. Without a rigid connection with the hand, it could, if it at all were moving, describe any other plane trajectory. Also then we would still consider that a force, a central force, is acting between the hand and consider it as the cause of its motion, its central motion. In order to be able to say anything about the size and direction of the force, we would have to know first the shape of its track and its velocity or its change of velocity, while it travels along it. Up to the present, we know the size and direction of a central force only during uniform motion along a circle.
Central motions, that is, motions of bodies about another body as centre without there being a visible connection are, for example, the motions of planets. Nikolaus Copernicus observed daily the motions of the stars about Earth; he explained motions of the planets with respect to each other and the Sun, the rise and setting of the stars above and below the horizon, etc., by a motion of Earth and the other planets about the Sun as centre (revolution). He considered the courses of the planets to be eccentric circles about the Sun. However, Johannes Kepler 1571-1630 discovered, based on more than 35 years of observations by Tycho Brahe 1546-1601 of the positions of Mars, that its trajectory is an ellipse, and therefore the Sun is located in one of its foci.
Kepler's first law (Shape of the planets' trajectories)
Kepler found, that the same is true for the other planets, namely: All planets move around the Sun in ellipses which have a common focus and in this common focus is the Sun. (Kepler's First Law)
The ellipse (Fig. 98) is a plane curve which, for example, is obtained by intersection of a circular cone with an inclined plane (Fig. 99). It is characterized by the relationship between its points and its two foci (S and S') which are located on its longest diameter, the large axis, at equal distances from its centre C. For each point (m, n, p · · ·) of the ellipse, the sum of its distances from the foci is the same.
The ellipse's oval shape shows that during its motion along an ellipse a planet changes its distance from the Sun S, that is, it is furthest from it at A (aphelion), closest to it at P (perihelion) (Fig.98). If a (=CA = CP) is half the longest axis and e(= CS = CS') the distance of its foci from the centre, but also the distance of the Sun S from C, the centre of the long axis, then the largest distance between the planet and the Sun is a + e, the smallest a - e, and the middle distance a, half the large axis.
The distance SC (= SC') of the foci from C, the linear eccentricity, determines the oval shape of an ellipse. The smaller it is in ratio to half the large axis, that is, the smaller e/a - the numerical eccentricity, the closer the ellipse resembles a circle and the smaller is also the difference between the largest distance a + e and the smallest distance a - e and the middle distance a of the planet from the Sun. Moreover the smaller is also the error, if one lets the planet travel around the Sun on a circle with radius a. The orbits of most planets deviate so little from a circle (Mercury most, Venus least) that one can use a circle except during astronomical computations.
The eccentricity of Mercury is 0,2056, of Venus 0.0068, of Earth 0.0167, whence the distance between the focus and centre of the ellipse is for Mercury 20%, for Venus 1.67%, for Earth 1.67% of half the large axis and the difference between the smallest and largest distances from the Sun, in per cent of the middle distance, for Mercury 40%, for Venus 1.2 %, for Earth 3.34%. The mean distances between the Sun and the planets are for Mercury 58, for Venus 108 and for Earth 149·10^{6} km.
Kepler's Second Law. (Velocities of the planets)
During his study of the problem whether Mars follows its orbit uniformly, that is, whether it covers in equal time intervals equally large arcs, Kepler discovered his Second Law, which he confirmed later on for all planets (Fig. 100): Let denote the arcs which a planet covers in equally long time intervals, then, by Kepler's Second Law, the sector SAB equals the sectors SCD, SEF, etc. In words: During equal time intervals, the radius vector - the line from the Sun to the planet - covers equal areas. This law answers the question whether the planet travels along the ellipse at uniform velocity. The form of the ellipse shows that, if SEF is to have the same area as SAB, must be larger than (because SE and SF are longer than SA and SB). Since the observed planet has passed along and in equal time intervals, it must have traveled along (when it is closer to the Sun) faster than along , that is, the planet travels along its orbit at non-uniform velocity and its velocity is the greater the closer it is to the Sun.
Mutual "attraction" of the Sun and planets
If you imagine the ellipses replaced by circles, the elliptic sectors are replaced by circular sectors. However, equally large circular sectors have the same arc length. You must then also assume that the planet describes in equal time intervals equal arcs, that is, the non-uniform velocities of the ellipse are replaced by uniform velocities along the circle. Compared with their distances, the planets and Sun are so small that they appear as points. (The diameter of the Sun is about 108th of the average distance of Earth from it and its volume is about 1.3·10^{6} times that of Earth.) The planet travelling along the fictitious circle around the Sun can be viewed as a material point, which travels at uniform velocity around another such point as centre of a circle - just as a point of a rotating rigid body travels about a circle around its axis. Hence we draw the same conclusions from the motion of planets along circles, that is:
1. The planet travels away from the centre tangentially
to its circular track ;
2. It cannot do so, because it is attracted by a centrally directed
force;
3. It opposes the centrally directed force
by an equal,oppositely directed force.
We do not know how it obtained its tangential motion. Kant 1755 and Laplace 1796 have attempted to answer this question by the nebular hypothesis regarding the creation of the planetary system.
The Nebular Hypothesis postulates for the members of the planetary system the same origin because of the similarity of their motions, the small eccentricities of their orbits and the minimal inclinations of their orbits to each other - of the planets as well as of their moons. The matter which now forms the planetary system was, according to this theory (Laplace) a lentil shaped nebula of very rarified gas, which extended initially as atmosphere of the Sun beyond the (present) orbit of the most distant planet. This nebula rotated about an axis which was perpendicular to the plane of the present planetary orbits. The gas cooled down from its surface inward, whence it became denser at the centre. As a consequence of its contraction, its angular velocity increased (according to the law of conservation of angular momentum and its outer sections separated from those closer to the axis as a ring ( Plateau's experiment). Further cooling down caused more contraction, yet larger angular velocities and separation of a second ring. The rings tore and their segments formed the planets, while the centre became the Sun. The hypothesis refers for its support to Saturn's rings and the creation of the Moon as a repetition of the formation of the solar system in the small. A formation, which differs essentially from the nebular theory, is considered today to be more likely. A main argument against it is the deviation of the axis of rotation of the Sun from the perpendicular to the plane of the rotations of the planets.
We also do not know how the force is transferred to the planet which stops it from following its inertia. But we must conclude^{1} that it is always directed towards the Sun, whence we see the Sun as the cause of the central motion of the planets, as the seat of an attracting force, which stops the planets from moving away from it. We must then seek in the Sun the point of attack of the force exerted by the planets on the centre (where the Sun is). That is, the central motion of the planets about the Sun suggests mutual attraction of the Sun and planets, which is equally strong and comparable to the centripetal and centrifugal forces. Newton gives the magnitude of the force F = 4p^{2}·ma/T^{2}, where T is the period of one revolution, m the mass, a (the mean distance, that is, half the large axis of the ellipse) the radius of the circular orbit of the planets.
^{1}This conclusion does not depend on the assumption that the planetary orbits are circles, for Newton proved them for the actual elliptic orbits.
Kepler's Third Law (period of planets). Magnitude of attraction of Sun
The magnitude of the force which the planet receives from the Sun depends therefore, apart from its mass m and its distance a from the Sun, apparently also on its period T. But only apparently on the period, the time of one circuit. If the planets were connected rigidly to the Sun, then all of them would have to complete their circuits in the same time T. However, they can move freely and have different periods; as astronomical observations show, the period is the longer, the larger is a planet's mean distance from the Sun, whence in the expression for F the time T differs from planet to planet.
With the masses m_{1} and m_{2}, the middle distances a_{1} and a_{2}, the periods T_{1} and T_{2 }of the two planets (1) and (2), the force on (1) is F_{1} = 4p^{2}m_{1}a_{1}/T_{1}^{2} and on (2) is F_{2} = 4p^{2}m_{2}a_{2}/T_{2}^{2} .
The ratio of the two forces is
Kepler discovered that the periods T and the middle distances were related like T_{1}^{2}/T_{2}^{2} = a_{2}^{3}/a_{1}^{3}, that is, the squares of the periods behave like the cubes of the middle distances from the Sun (Kepler's Third Law). For example, you have the equation
(T_{Earth}/T_{Venus})^{2} = (a_{Earth}/a_{Venus})^{3}.
Hence you can replace T_{2}^{2}/T_{1}^{2} by a_{2}^{3}/a_{1}^{3}, and thus find F_{1}/F_{2} = m_{1}/m_{2}·a_{2}^{2}/a_{1}^{2}, indepedently from the periods T_{1}^{ }and T_{2}. If you write this equation in the form
F_{1}/(m_{1}/a^{2}_{1}) = F_{2}/(m_{2}/a^{2}_{2})
and take into consideration that (1) and (2) are two arbitrary planets, you see that the same equation holds also for two other arbitrary planets, that is, F/(m/a^{2}) has the same magnitude for all planets. Call this quantity m, so that
Centripetal force of the Sun/[(Mass of the planet)/(Square of its mean distance)] = F/(m/a^{2}) = m,
that is, F = m/a^{2}·m, where m has the same value for all planets. If there existed a planet with mass m = 1 at the distance a = 1 from the Sun (conceived as a point), then F = m , that is, m is the magnitude of the force by which the unit of mass is attracted, if it is located at one length unit from the Sun. However, the masses m differ and so do the distances a. For masses which are 2, 3, · · · times as large as F, by the above equation, 2, 3, · · · times as large as for the mass 1 and its magnitude at 2, 3, · · · times the distance 1/4, 1/9, · · · 1/a^{2 }times as large as the force at unit distance. In other words: The attracting force of the Sun is proportional to the mass of the planet and inversely proportional to the square of its distance from the Sun.
Also the motions of the Moon about Earth, of Jupiter's Moons about Jupiter, etc., are central motions. Newton formulated therefore the hypothesis that the central motions of all stars are the result of one and the same force, that the force exerted by Earth on the Moon is of the same kind as that exerted by the Sun on the planets, that the same force which drags the Moon (away from the tangent to the Moon's orbit) towards Earth is of the same kind which makes bodies near Earth's surface drop to Earth, as well as the weight of a body on Earth and the central motions of the stars. They are confirmations of one and the same force.
Kepler's laws only refer to the relations of the planets to the Sun. However, every planet is also attracted by all other planets. Certainly, the attraction of the Sun by far outweighs due to its large mass the interaction with the other planets, a fact which enabled Kepler to formulate his laws. However, these laws can only be approximations to reality. And indeed over many years . the interactions become noticeable as deviations (perturbations) from these laws. Among these deviations, one has a slow rotation of the axes of the planetary orbits (in their planes) and thereby of their perihelia relative to the system of fixed stars. The observed motions of the perihelia are in agreement with those computed by means of perturbation theory for all the larger planets; in contrast, the computations for Mercury yield a value which is 43" per century too small. Einstein's gravitation theory explains the hitherto unexplained amount by the action of solar gravitation without a need for other hypotheses. In the case of Mercury, the deviation from Newton's theory can be measured, but not for the planets which are further away from the Sun; the deviation decreases strongly with increasing distance of the planet from the Sun and would be undetectable already at Earth's distance. The orbit of Venus differs so little from a circle so that the position of the perihelion is only vaguely known.
The Moon is attracted by Earth - why does it not fall onto it? According to Newton, it does fall, but 1/60^{2 }=^{ }1/3600 as fast as, for example, an apple drops from a tree, because it is 60 times as far from Earth's centre - 60 Earth's radii - while the apple is only one away. Thus, the Moon is only attracted 1/3600 times as an equally large mass on the surface of Earth.
Denote by M the mass of the Moon, by v its (assumed to be uniform) velocity along its (assumed to be circular) orbit about Earth, r the radius of its orbit, then the force F by which Earth deviates it from the tangential direction, in order to keep it on its orbit, is F = M·v^{2}/r. The force by which a mass equal to the mass M of the Moon is attracted on Earth's surface is its weight P = M·g. If Newton's idea holds, you must have approximately P = 3600·F. Using the known numerical values for r, g and v in the equation P/F = r·g/v^{2}, whence it is seen to be correct.
Thus, the Moon falls towards Earth, although only 1/3600-th as fast as a body near its surface - in the first hour (3600 seconds) only as far as the body in the first minute. Why does the Moon not come closer to us? Because beside the motion towards Earth it executes a motion at right angle to it like the stone flung around by means of a rope (Fig. 94) which is attracted to the hand, but nevertheless does not approach it, because it moves following its inertia perpendicularly to the direction in which it is pulled. If you place in Fig. 97 Earth at C , assume the circle to be the Moon's orbit, replace the rope in Fig. 94 by the gravitational attraction, you will understand why , although it moves along a radius of its orbit towards Earth, the Moon does not approach Earth: It moves ("falls") all the time away from the line, to which it is moved by its inertia, away to the centre of Earth; but it starts effectively every instant to fall again along another radius of its orbit, its fall causes it only to strive away perpendicularly to its orbit's changing tangent in order to remain on the circle. If it were not to fall, it would not maintain its distance from Earth; in contrast, its inertia would carry it away along the tangent (at the changing point on its circular orbit); it maintains at all times its distance from Earth, because it falls all the time with the same acceleration towards it.