To enunciate this fundamental principle of Nature it is necessary to define the torque
of a force and the total torque of a system of several forces.
If we consider the simple example of a bar which is revolving in a vertical plane around an horizontal axis passing across one of its extremes, and we take in account that, if the bar is homogeneous and is made wholly of the same material, its weight force can be considered as it were applied in its barycentre, that is in its middle point, then we can verify that the state of stable equilibrium of the bar can be achieved only when it is disposed vertically.
It is,in fact, nothing but a physical (not ideal) pendulum that, when it isn't oscillating, reaches the stable equilibrium position, only when it is disposed on a vertical line, with its barycenter placed under the hanging point.
In fact, only in this position the weight force has no moment around the rotation axis; consequently no rotation happens.
Therefore the bar, if it isn't oscillating and is placed in its stable equilibrium position, remains in it indefinitely, till it isn't made to oscillate by applying to it, in a shortest time, an impulsive force having a torque around the rotation axis.
The torque of the weight force around the rotation axis is non-zero only if the so-called lever arm of the force is non-zero.
The lever arm of the force is the distance between its action straight line (the direction of the force) and the hanging point of the bar.
The torque of a force applied to a rotating body is a vector directed, by convention, along the rotation axis and its modulus is equal to the product between the modulus of the force and its lever arm around the axis.
If we consider,for example, the applied force has the intensity F = 1 kg ~= 9,8 N (newton) and its line of action is distant b (lever arm) = 10 cm = 0, 1 m from the rotation axis, the torque M of the force F is gotten by multiplying the intensity of the force by its lever arm b = 0,1 m:
M = Fb = 1 x 0,1 = 0,1 kg x meters, or: M = Fb= 9, 8 x 0, 1 =
= 0,98 N.m (newton x meters).
When the bar oscillates, the torque of the weight force P around the rotation axis is given by the product M = P b of the weight force P by the lever arm b, that, in this case, is the distance between the rotation axis and the vertical straight line passing across the barycenter of the bar.
The torque value of the weight force is proportional to the aforesaid distance, is zero when the barycenter is placed under the rotation axis, because in this case the lever arm is zero, and changes sign when the bar goes beyond the stable equilibrium position.
Therefore the bar, after a complete oscillation, comes back to the stable equilibrium position and then goes beyond it, continuing to oscillate till the friction force doesn't dissipate the whole initial kinetic energy.
If to a rigid body,provided it is free to rotate around a fixed axis, we apply several forces, the total torque (or the resultant torque) acting on the body is gotten by adding all the torques produced by the forces making the body rotate clockwise and subtracting all the torques produced by the forces making the body rotate anti-clockwise.
In the rotatory motion the torque behaves as the force in the translational motion (the motion of a body that doesn't revolve around its barycenter), because it produces an accelerated rotatory motion, with an angular acceleration around the rotation axis.
Practically, to express the rotatory effect (the resultant torque) of a system of several forces, independently from their lever arms around the rotation axis, it is considered the torque of two parallel forces, having the same modulus and opposite directions, that are equivalent to the force system.
The torque of a couple is definite without considering the lever arms of each force around the rotation axis, but by calculating the product of their moduli by the distance between their action lines, that is the lever arm of the couple.
If it is known that the engine torque of any motor has the value of 100 kg.meters, it
means that both the antiparallel forces that constitute the couple have each the modulus
of 50 kg.meters and the lever arm of 2 m, or the modulus of 500 kg and the lever arm of
0,2 m = 20 cm, or also that they produce a torque like the one of any pair of factors
forces x lever-arm giving the product of 100 kg.meters .
In an equivalent mode one may say that the torque of the engine couple of the motor can be balanced by the torque of a force of 1000 kg applied tangentially to a distance of 0,1 m = 10 cm from the rotation axis, for example by using a pulley with the radius of 10 cm, furnished with a cable to apply a force that could balance the effect of the engine couple, so that the rotation of the engine shaft is stopped .
In alternative, the torque of the engine could be balanced by the moment of a force of 100 kg, applied tangentially by a cable to a pulley with the radius of 1 meter.
In other words, there is in mechanics a well known rule (the so-called gold rule), which derives from the Archimedes theory of the lever: the force F that must be applied to a body rotating around an axis to get a resistant torque R that is able to balance a engine torque C, varies with the inverse proportionality law in terms of the lever arm b:
R = C = F b ; F = R/b.
The gold rule states that the smaller is the intensity of the force F, the greater is the distance ( lever arm b ) of the application point of the force from the axis of rotation.
If we consider that the length s of the circumference arc described by the application point of the force F, for a rotation of n radiants, is directly proportional to the lever arm b, that in this case is equal to the radius r of the pulley ( s = r n ), we realize that the mechanical work W that has to be made to balance the effect of the engine torque C is
W = F s = F r n , and that therefore, because the displacement s varies with the inverse proportionality law with respect to the applied force F, then the advantage that is gotten with the reduction of the modulus of the force F that has to be applied, is paid with the disadvantage of the greater displacement s.
An analogous gold rule is worth for the gear of a car, whose structure derives from the improvement of the speed change device invented by Leonardo da Vinci.
The gear of a car is a converter of the engine torque, that makes it possible to use the mechanical power generated by the engine in the more suitable conditions, in relation to the mechanical load requested in the various driving conditions: starting, starting on a sloping road , reversing, going with low mechanichal load and high speed on a rectilinear road).
If P is the power the engine gives to the engine wheels of a car,the gear, according to the driving conditions, transmits the greatest torque Cmax (the greatest couple) with the least angular velocity Vmin ( P = Cmax Vmin ), or the least torque Cmin (the least couple) with the greatest angular velocity Vmax ( P = Cmin Vmax).
The gold rule in this case states that the advantage of disposing of a great torque (great couple), is paid with a smaller speed of the vehicle.
When a low march (I,II,III) is used, by a system of gears is gotten a reduction of the number N1 of the revolutions per minute ( directly proportional to V ) and the greater torque C1, requested to win the great motion-resistance forces, with a given power (P), is supplied by the engine.
When a high march ( IV, V ) is used, the smaller torque C1 is gotten by increasing the number N2 of the revolutions per minute, with the same power supplied by the engine: C1 N1 = C2 N2 = P .
In the rotatory motion the physical quantity that determines the aptitude of a rigid material system to withstand to the accelerating action of a torque is the momentum of inertia J (rotational inertia), that depends on both the mass distribution around the rotation axis and the squared distances among the masses of the system and the fixed rotation axis.
If, for example, we consider a metallic bar with mass M and length L, free to rotate about a fixed axis, if the axis is passing across the middle point of the bar, the momentum of inertia is J = ( 1/12 ) ML2 .
If instead the rotation axis is passing across an extreme, the moment of inertia is 4 times greater: J = ( 1/3 ) ML2 , because the most mass is placed at a greater distance from the rotation axis, and that implicates a greater rotational inertia.
In the rotatory motion the physical quantity which is analogous to the linear momentum of a body in translational motion, is the angular momentum L or the torque of the linear momentum, that is gotten considering the sum of the products between the generic momentum
Pi = MiVi of one of the n masses of a material system and the lever arms bi around the rotation axis:
L = P 1b1 + P2 b2+..Pn bn = M1V1 b1 + M2 V2 b2 + ..Mn Vn bn.
In the case of a hard body free to rotate around a fixed axis, the angular momentum L is gotten by multiplying the moment of inertia J (expressed in kg.m2 ) by the angular velocity
w = 2p N/60 ( expressed in radiants/second), where N is the number of revolutions/minute:
L = Jw.
By analogy with the second law of dynamics (Galileo-Newton's law) F=dP/dt = M dV/dt),we can write the second law of dynamics of the hard bodies rotating around a fixed axis: M = dL/dt =J dw /dt = J a, where a is the angular acceleration of the rotatory motion.
If in particular, the torque M of the external forces is constant, we got a rotatory motion with a constant acceleration, for which the variation of the angular momentum in the time t is directly proportional to the resultant torque M of the external forces that accelerate the hard body:
M t = L final- L initial = J ( w final - w initial) .
The study of the precessional motion of a top is studied by applying the second
law of the rotational dynamics.
If we make a top rotate very fast, we notice that, when its rotation speed decreases, because of the aerodynamic resistance and the friction forces acting between the peak of the top and the support plane, the inclination of the top with respect to the vertical is more and more increasing, while describing the side surface of a cone, the so-called precession cone.
The precession of the axis of a top is analogous to the precessional motion of the rotation axis of the Earth, that is inclined of 23° 27' with respect to the perpendicular to the orbit plane (the so-called ecliptic plane ) and completes a revolution in about 26000 years (the platonic year ), determining the precession of the equinoxes,that consists in the systematic advance of the points of the equinoxes in comparison with the ones of the preceding year.
In the case of the top the precession of the rotation axis, after some time ,that is the greater, the greater is the initial rotation speed, happens the instability of the top motion , that is produced by the torque of the weight force applied to the top barycenter, around the contact point between the top and the ground.
This perturbing torque generates an additive angular momentum that acts along the perpendicular to the top rotation axis and is added vectorially to the main angular momentum, around the rotation axis.
Therefore the top axis is made to describe the side surface of a precession cone, which is the larger, the lower is the rotation speed of top, till this leaves off revolving.
As regards the Earth, the precessional motion is determined by the gravitational forces produced by the Moon and the Sun on the Equator zone of the Earth, that,as we know, is a little flattened at the poles.
In a material system,according to the third principle of dynamics,the torques of the
inside forces are balanced two by two each other,
because they are action and reaction forces that produce equal and opposite torques.
Therefore, if the resultant torque M of the external forces is zero, the variation of the angular momentum per second is dLtotal/dt = M = 0, then the total angular momentum remains constant, that is it keeps over:
L total = L1 + L2 + L3 + ... Ln = constant (that is the conservation principle of the angular momentum ).
If we consider, for example, a planet revolving around the Sun,provided we neglect, to
simplify the problem, the gravitational forces effected by the other planets, we realize
that, because the gravitational attraction forces inside the Sun-planet system, are equal
and acting in opposite directions, the total angular momentum L = MVb of
the planet along its elliptic orbit mantains constant.
Because b is the distance between the tangent line to the orbit and the Sun, that is placed in one of the focuses of the ellipse, and the velocity V of the planet is directed, in each point, along the tangent line to the orbit in the same point, the modulus of V varies with the inverse proportionality law in terms of b, by decreasing with the increasing Sun-planet distance.
Therefore the orbital speed of the planet is least near the aphelion (the point of maximal distance from the Sun ) and instead it is maximal near the perihelion (the point of least distance from the Sun ).
From the angular momentum conservation of the planet we deduce that the area described from the straight line from the planet to the Sun is directly proportional to the time (the II law of Keplero,that is the law of the areas).
The conservation of the angular momentum is applied to the rotating air masses that
generate the vortexes in league of cyclones, tornados and water-spouts.
In fact, since the only external force acting on a rotating air mass is the gravity, which is directed along the vertical, there is no torque around the vertical axis of the rotating air mass and consequently the angular momentum preserves.
Therefore,if an air mass spinning with a low speed is pushed from the pressure difference ( baric gradient ) toward the center of the cyclonic area, its angular velocity increases rapidly approaching to the center of the depression, that is to the cyclone eye (the center of the vortex).
Analogous phenomena happen in the whirlwinds and in the water-spouts trumpets, that, because of the greatest depression at the center of the vortex, suck everything is placed along their destruction trajectory.
The conservation principle of the angular momentum explains, in particular, the generation of the vortex we observe when taking off the plug of a bath-tub.
The tangential speed V of the liquid mass M at the distance R from the orifice, increases with the inverse proportionality law with the decreasing distance R: Vo Ro = VR.
In fact, since the gravity force has no torque in comparison with the orifice, the angular momentum preserves: M Vo Ro = M V R.
On the angular momentum conservation is based the operation of a gyroscope and of a
gyroscopic compass, that consist of a thick metallic disk which is spinning with high
speed around an axis mounted with two least-friction pivots on a Cardan hanging frame.
Since the friction is negligible and the force of gravity has no torque around the Cardan hanging point, which is placed in the barycenter of the spinning disk, the angular momentum of the disk preserves, and,because it is a vectorial physical quantity, the rotation axis of the disk preserves its initial orientation.
The operation of bicycles and motorcycles founds on the conservation principle of the
angular momentum .
In fact, if the driver makes the gravity force, applied to the barycenter of the system driver-vehicle, have no torque (this is the equilibrium condition) around the contact points among the wheels and the road, the angular momentum of the system preserves nearly constant both in value and direction; consequently the wheels rotate in a nearly vertical plane.
We can say that, because of the friction forces and the variability of the engine torque applied to the wheels in relation to the road conditions, the varying speed and the angular momentum of the wheels make the inclination of their plane be departing from the vertical, implicating variations of the wheels in comparison with the vertical plane.
The conservation principle of the angular momentum is essential to explain many other
dynamic phenomena, that are, for example, the pirouettes of ballerinas and gymnasts.
A ballerina, pivoting on a foot and contracting or stretching the superior and inferior limbs,is able to revolve around the vertical with an increasing or decreasing angular velocity.
In fact, the contraction of the limbs toward the vertical rotation axis, makes the momentum of inertia J decrease with the increasing mass distribution near the rotation axis.
Then, by means of the nearly constant angular momentum L =Jw, the angular velocity w of the ballerina increases.
Instead, the angular velocity decreases,when the ballerina stretches the limbs.
Generally, if a system consists of any bodies both revolving around their barycenter and having also an intrinsic (rotation) motion around itself, provided the total torque of the external forces acting on the system is zero, preserves the total angular momentum, given by the vectorial sum of all the rotation intrinsic angular momenta and the revolution angular momenta around the barycenter.
In this case, not only any variations of the revolution angular momenta of the bodies around the barycenter may happen, but also any conversions of their intrinsic angular momenta into revolution angular momenta around the barycenter, and viceversa, provided the vectorial sum of all the angular momenta (both of rotation and revolution) preserve .
By means of the conservation principle of the angular momentum we can explain the
increasing distance of the Moon from the Earth because of the friction produced by the
In fact the effect of the lunar gravitational attraction on the seas determines an oval deformation of the equilibrium surface of the oceans, that follows the orbital motion of the Moon, that is about 27,5 times slower than the Earth rotation about its axis.
The consequent friction among the sea waters and the Earth's crust determines a continuous decreasing of the angular velocity,the kinetic energy and the rotational angular momentum of the Earth, and a consequent increasing of its period of rotation (day) of about 16 tenths-thousandthes of second per century.
The decreasing rotation angular momentum of the Earth, in consequence of the
conservation principle of the total angular momentum of the Earth-Moon system, is
compensated by an increasing orbital angular momentum of the Moon, provided we consider
are negligible, in first approximation,both the variation of the rotation angular momentum
of the Moon around its axis and the average value of the resultant torque of all the
external forces effected by the Sun and the other planets.
The orbital angular momentum of the Moon is given from the formula:
L Moon = M Moon x V1 Moon x R1 Earth-Moon = M Moon x R2 Earth-Moon x V2 Moon orbit, where w = 2 p/T Moon ~= 2p/ ( 27,5 days * 24 hours * 3600 seconds ) is the orbital angular velocity of the Moon.
Therefore, because L is increasing, also R (the Earth-Moon distance) is increasing much slowly.
The mirrors place0d on the Moon surface by the astronauts of the Apollo tasks made it possible to determine, by a laser beam, that the Moon is leaving from the Earth of about one centimeter per year.
The conservation principle of angular momentum has a fundamental importance also in the microcosm, in spite of the Galileo-Newton mechanics laws cannot be applied to the subatomic elementary particles, that follow instead the quantum mechanics principles.
In fact, in the atomic, nuclear and subnuclear physics and in the quantum field theory , besides the rotation angular momentum
(orbital ),for example the orbital motion of electrons around the atomic nucleous, also the intrinsic angular momentum of the particles with spin must be considered, that is the rotation (intrinsic) angular momentum of a particle around itself, even if the principles of quantum mechanics prevent that both the spin and orbital angular momenta assume any value, but only certain values, according with the quantization of the physical quantities in microcosm systems.