A6. General Theory of Motion and Force (Mechanics)

Effect of Earth's rotation on the direction of fall

In spite of its special significance, we will refer here to the exceptional case of a horizontal throw. A freely falling body really does not fall under the conditions specified. We excluded there Earth's rotation, although it always acts. Earth turns from West over South to East and therefore gives every body, which is tightly linked to it, a velocity from West to East. A body which drops freely from a tower is therefore strictly speaking propelled Eastward. The tangential velocity at the top of the tower is somewhat larger than at its base below, whence Earth's rotation deflects the body Eastwards (away from the base). It does not arrive at the bottom end of the vertical through the point of propulsion, but to the East of it.

The experimental problems, first of all the difficulty of determining the point of the vertical and then to launch the body without perturbations, are enormous, since the deflection for 80 m is only about 1cm. A comparison of the experiments of Hall 1829-1907 1902 with a fall height of 23 m and of Kamille Flamarion 1842-1925 1903 with a height of 68 m, which involved mean errors of 3.3 % and 22%, respectively, has shown that smaller heights yield better results. According to the theory, the Easterly deviation is proportional to the product of height and duration. In order to achieve measurable results, in spite of the smaller height, the duration must be made as long as possible.

I.G. Hagen 1847-1930 1912 examined therefore the effect of Earth's rotation on a falling body in Atwood's machine. He discovered for the height of 23 m and a fall acceleration of about 1/25 g a deflection of 0.9 mm, a result which agrees with the theory to within 1%. - Earth's rotation also diverts bullets from their vertical trajectory (Coriolis deviation).

Composition and decomposition of forces

Hitherto we have only discussed simultaneous forces and motions. If a material point is subject to three simultaneous forces (Fig. 22 a, b, c), the third force c combines with the resultant R1 of the first two forces a and b into the resultant R1 of a and b into a new resultant R2. Similarly, a fourth force combines with R2, etc. (Force polygon) Conversely, you can imagine decomposition of a force, which is represented by the straight segment AB (Fig. 23), to have arisen by decomposition of the forces AC and AD, or equally well out of AE and AF, or, more general, out of any two forces with which you can construct a force paralellogran with AB as diagonal. Since every component can be viewed as a resultant, every force can be decomposed and composed into others by means of the parallelogram of forces.

 

 

 

 

 

 

 

Assuming that n forces act on a material point and (n - 1) have a resultant, which is exactly as large as the last of the n forces and has the opposite direction, then the n forces are the same as two equally large, but oppositely directed forces. Two such forces balance each other, because they move the material point in opposite directions with the same acceleration, and are the same as if no force acts. The state of motion of the point has therefore not been changed by these n forces, it is at rest; if it moves along a straight line with uniform velocity, this also does not change. We see that under certain conditions, in spite of the action of forces, a point's state of motion remains unchanged.

Vector. Scalar

The parallelogram of forces (and its generalization) yields the basic scheme, by which directed quantities, vector quantities, are composed and decomposed. A vector is a quantity which is completely characterized by a number - its measure - and a direction: Velocity, acceleration, force are vectors - their magnitudes are so many units and they are directed in some direction. They are represented by arrows, the length of which is their measure and the pointed end of which indicates their direction. The word vector arose out of the simplest of the directed quantities, the radius vector, a segment drawn from a fixed point to a movable point. You call such an arrow a polar vector (in contrast to an axial vector). Quantities which are not associated with directions like temperature, mass, density are already fully determined by a number and a unit. They are called Scalars (Hamilton 1853), because their value, measured by means of a scale subdivided into certain units, is fully determined.

Geometrical composition and decomposition of vector quantities are tools, used daily in Physics, because natural processes turn out to be composite. They have a dominant role in all of Physics, for example, in Electricity, where the effect of several electrical fields or magnetic fields are added in vector diagrams. The scheme of the geometrical addition of two vectors like 1 and 2 in Fig. 25 is always as follows: Place at the end of Vector 1 the start of Vector 2 with its direction; Vector 3, which closes the triangle from the end point of Vector 2 to the starting point of Vector 1 is the required vector in the opposite direction. The figure shows the application of this scheme to the composition of two forces or motions (with the same result as is obtained by the parallelogram approach).

If you add several vectors, you obtain an open polygon in space, along which you move in the directions of the individual vectors.The side which closes the polygon yields then the desired vector with the same magnitude, but opposite direction. Fig. 26 shows the addition of three (not in equilibrium) forces; it yields the same result as the parallelogram of Fig. 22. The parallelogram for the equilibrium of three forces, acting at one point, shows that the three vectors close on addition the triangle, that is, that they balance each other.

Conversely: If a material point is at rest or moves along a straight line uniformly, this does not yet imply that no foce acts on it, but only that the forces acting on it balance their actions. Each of the weights in Fig. 24 tries to move the point P in another direction. Nevertheless, P remains at rest, and indeed because, as the construction of the parallelogram of the forces would display, the resultant of the forces A and B is balanced by C. In this case, you say about the forces: They hold mutually in equilibrium. (A railway train moves, driven by the enduringly acting force of the steam engine, with uniform velocity; it is not accelerated. The forces due to friction are the (n - 1)components, the force of the engine is the n-th component. A change of its state of motion is impossible, because there is equilibrium of the forces.) Conversely: If a material point is subject to forces, but is nevertheless to maintain its state of motion unchanged, this can only happen if yet one more force is added, which annuls their resultant, that is, is equally large to it and oppositely directed, that is, it reestablishes equilibrium.

Equilibrium of forces. Statics. Dynamics

The theory of the equilibrium of forces is called Statics, that of the motion with special consideration of the forces which cause it Dynamics.

Experience tells you this with all gadgets which allow you to overcome forces by other forces, especially when your own muscles are involved. If you displace a body as a whole, for example, lift or pull it, you sense exertion; similarly, if you deform a body (saw, press, bend, etc.). irrespectively of what gadget you investigate, you find everywhere two groups of forces face to face, and that the goal of all mechanical gadgets are change of the position of a body. This also includes relative displacement of parts of a body as, for example, during bending or other deformations. The parts of a deformed body are a set of mass particles which after deformation are in different relative positions to each other.

One of these two sets of forces aims to cause a certain change of relative positions, in the above example your muscles. The other resists it - in the examples above, it is gravity, friction, inertia, strength and other forces active between the parts of a body which constitute the essence of cohesion. In order to achieve an intended deformation, you must balance the resisting forces. If that has happened, the respective body is such as if no forces act on it.

For example, consider the interaction of the forces on a railway train, which is to be moved and maintained at a certain velocity. Let the objective be uniform and, for the sake of simplicity, motion along a straight line. The force which is to be employed to achieve the intended change of position is that of the engine. If no force acted on the train, a minute force would suffice, in order to move it and gradually give it the intended speed. Once this has been achieved, no further force would be required, because the train has achieved its velocity and inertia would keep it going. In reality, the train is subjected to forces other than that of its engine, all of which resist its motion: Friction of its wheels on the rails and in its axles, the air, etc. form a system of forces with a resultant, which has a certain magnitude and direction and impedes the mobility of the train. Its action is balanced by a force of equal magnitude and opposite direction. Already a minute excess of the force supplied by the engine suffices to accelerate the train by prolongation of this action and allows it eventually to attain the intended velocity. Once this velocity has been reached, the engine's force can be reduced and only two forces, which balance each other, will act on it . The velocity is maintained through inertia which acts without hindrance.

What was true for the engine's force and the train, you can note on yourself as you push a load and maintain its motion, for example, when you push something ahead of you or pull behind you; you must exert yourself more at the start of the motion (to overcome inertia, friction) than to maintain it (overcome only the friction). The same cause makes it more exerting for a horse to set a cart into motion than to keep it going.

Work

As in all mechanical arrangements, here too occur changes of position and two pairs of forces face each other, one of which resists the change of location. This set of forces is called resistance or load; the process of overcoming them is called work. "Work is the act of causing a change in the configuration of a system against a force which resists this change". (Maxwell 1876 "Matter and Motion"). Strictly speaking, you can speak of overcoming only when the one party is stronger than the other. In contrast, in common language, if equilibrium occurs neither of the forces is the overcoming and/or the overcome one. The force impedes the resistance from lowering the velocity already attained, the resistance stops the force from increasing the velocity. The work, performed by the force, after the motion has become an inertia motion, is that of balancing the resistance.

The physical concept of work has been taken from the work of Man and animals. Its generalized significance in Physics and the details, according to which its magnitude may be assessed, are readily understood when you think of work performed by your muscles.For example, if you lift a weight from the ground, you perform work. The force involved is that of your muscles; it attempts to take the mass away from Earth and is directed vertically upwards. The resisting force, the attraction of Earth, acts vertically downwards. This force is familiar to you as the weight of the mass. (In order to avoid confusion, we will denote the weight of one kilogram by 1 kg* and its mass by 1 kg (correspondingly, g*, mg), etc.)

The magnitude of the force required to maintain a mass in uniform, vertical, upward motion, must be equal to that by which Earth attracts it. Hence, if you want to move 1 kg mass in such a way, you must let a force act on it which acts upwards and equals the weight which pulls it downwards, that is, a force of the magnitude of the weight of 1 kg (1000·g dyn, note that here g does not denote gram, but the gravitational acceleration). Similarly, in order to move m kg mass at uniform velocity vertically upwards, you require the force of m kg*.

Unit of work (meter-kilogram)

By lifting 1 kg mass by 1 m you have performed a certain amount of work. If you lift it by another m, a third, etc., you must perform again the same work for each m; in order to lift it by h m, you must perform h times as much work as was required for 1 m. (Gravity resists lifting by each metre by the same amount under the presently considered possibilities over Earth's surface.) Stated briefly, the work performed during lifting of a mass is proportional to the height, by which the mass is lifted, or it is proportional to the length of the path along which the load is overcome. The work required to lift 1 kg* over 1 m is called a metre-kilogram (1 mkg*).

So far, we have only talked about a single kilogram. In order to lift a second kg*, a third kg*, etc. to the same height, that means to perform again the same work for every kilogram, for the performance of p kg* (that is, to overcome p times that resistance). Thus: The magnitude of the work, performed during lifting, is also proportional to the weight of the lifted mass, that is, proportional to the force which puts up the resistance. We have set the work required to lift 1 kg by 1 m equal to 1, in fact, = 1 mkg*, that is, we have defined the metre kilogram* as the unit for the magnitude of work, whence the work p mkg* is required to lift p kg by 1 m. Work is equal to the magnitude of the resistance, which must be overcome, multiplied by the distance, over which it must be overcome.

We have talked here only about the resistance and the distance, along which it is overcome, that is, we have looked at it from the side of the resistance. However, you can also look at it from the side of the working force; it is equal to the resisting force. Displacement of a mass by h m during work implies a shift of the point of attack of the working force by h m. Hence we can define the magnitude of work: The magnitude of work equals the product of the working force by the distance by which the force's point of attack is shifted.

Erg. Horse power

The definition of the metre-kilogram is not exact: The weight 1 kg* differs with its location on Earth. For technical applications, this difference is unimportant, but not for Physics. It has therefore been defined as follows: The unit of work is the work, performed by 1 unit of force (1 dyn) during shifting of its point of attack by 1 unit of length (1 cm). This unit is called 1 erg.

The unit of force is approximately the weight of 1 mg, whence 1 erg is approximately the work performed by lifting 1 mg by 1 cm. - Since 1 g* = 980 dyn at a location, where the gravitational acceleration is 980 cm/sec², then at this location 1 kg* = 1000·980 = 98·104 dyn, and since 1 m = 100 cm, 1 mkg* = 98·106 erg.

Work, done during the time unit, is called effect - its unit is 1 erg/sec. The work 107 erg is called 1 Joule, the effect 107 erg/sec 1 Watt. Engineers measure by effect in terms of Horse Power (hp) equal to 75 mkg*/sec, that is 735·107 erg/sec = 735 Watt.

A Warning: We have used lifting of a mass as the example of work, because everyone knows about it. However, do not consider it to be a special kind of work. Lifting of a mass means nothing else but overcoming of a force by which the mass is attracted in a given direction. In this example, the force happens to act vertically downwards, and that is why the force of the muscles is exerted vertically upwards. However, this does not differ from, for example, a force trying to drive a body northward and us using our muscles to pull it southward. For the magnitude of work done, only the magnitude of the resisting force and the distance, by which the mass is shifted by the performing force, is important.

The magnitude of the performed work always depends on the magnitude of the resisting force and the distance over which the mass is displaced. The performed work always equals the product of the magnitude of the force and the displacement; the force must be given in dyn and the displacement in cm, in order to obtain the work in erg. Since 100 000·g erg = 1 mkg*, every work, which you know in erg, can be converted into mkg*; in this way you can visualize how many kg this work, for example, could raise by 1 m.

Energy. Kinetic energy. Potential energy

We have done work. What have we received in return? At the start of the work, the mass was on the ground, during the work it rose, and after completion of the work. the body is located at a certain distance above the ground. The result of the work is the new location of the mass relative to the ground. Since it is above the ground, it is in a position to fall. Falling masses (water, weight) are known to be able to perform work. By the fact that the mass is above the ground, it has gained the ability to perform work.

This ability is called Energy ( Young). We have exchanged our work for the energy of the lifted mass, that is, its ability to perform work. But it can perform work only as it falls. A rammer, however heavy or high up above the ground it may be, does not perform any work, if it does not fall. Thus, the ability to perform work has the mass in the first and second state, but it must change from the state of rest to one of motion, in order to realize its potential to perform work. In the first state, the mass can be compared with a reservoir of work, but a reservoir the content of which can only be used in the second state.

Not only falling, but every somehow moving mass is known to be able to perform work: Moving air (wind or storm), a flying bullet, streaming water, a moving railway train, etc. The ability to work which a mass has through its being in motion is called kinetic energy. The energy which it possesses as a result of its location, like the mass which has been lifted above the ground and is able to drop, is called potential energy.

How does it happen that the motion of a mass realizes its ability to work? You must understand that motion of a mass is always the result of work performed. When we talked earlier of the work of a mass, the force, although it acted lastingly, maintained only its uniform motion. An acceleration was impeded by the performing force being opposed to another force, a resistance. The work arose because another force resisted. If that other force does not exist or suddenly ceases to act, naturally the performing force causes acceleration - can you still then talk of an effect of the moving force?

Yes! Due to its inertia, the mass aims to maintain its instantaneous state of motion. If it is to be accelerated, its inertia (its resistance to any change of its state of motion) must be overcome at each point of its track, that is, work must be performed. Also, during this work, the mass is moved over a certain distance. Let p be the work performing force, h the displacement of the mass in the direction of this force, then the magnitude of the work performed by the force is p·h (in erg, if p is in dyn, h in cm.)

The effect of this work p·h is that the mass m has been displaced by the distance h cm with acceleration. When the force started to act, let m have the velocity 0; when it had been displaced by h, let it have the velocity v and have taken t sec to get there. In order to compute the work required under these conditions to displace m by the distance h, the force and the distance h must be expressed in terms of already known quantities.

We know already that the distance, covered during the time t, that is while the velocity rose uniformly from 0 to v, is as large as if m had moved during that time with the velocity v/2, that is, the distance equals t·v/2. Moreover, we know that the velocity has risen uniformly in t seconds by the velocity v, that is, in 1 sec by v/t, so that the acceleration is v/t; the force, which acted on m, is therefore m·v/t. The work required to produce this increase in the velocity is

force·path = m(v/t)·v(t/2) = mv²/2.

We have already denoted this work by p·h, whence

p·h = mv²/2.

mv²/2 is called the kinetic energy of the mass m.

Principle of kinetic energy. Conservation of kinetic energy

We have asked: What have we received in exchange for our work? Prior to the start of the work, the mass had the velocity 0, at the end of it the velocity v. The increase in its velocity corresponds to the amount of work done. Hence we can view the final velocity to be the result of the work. Assuming the force p, which gave the mass m the velocity v, is opposed by a force q. It would, if m were not in motion, give it motion in its direction. However, since m is moving, it can do no more as to retard this motion; however, at first the mass will continue to move in its old direction. The counter force q represents a resistance which the moving mass overcomes. Gradually, its velocity becomes zero, and then, and only then, its capability to do work is exhausted. The work, which the force p did on the mass m and which had been in a sense stored at the magnitude mv²/2, becomes exhausted.

The expression mv²/2 indicates the magnitude of the work which the mass m can perform thanks to its velocity. For example, you throw the mass m with the velocity v vertically upwards. It has initially the kinetic energy mv²/2. It rises to the height v²/2g and overcomes gravity underway. The force which pulls it downwards is its weight mg. Thus, the mass performs, thanks to its motion until its velocity becomes zero, the work: Path·force = (v²/2g)·mg = mv²/2. Its initial kinetic energy is therefore as large as the work which it can perform due to its motion.

This work, performed by motion while the velocity fell from v to 0, is as large as the work, which must be done, in order to give m the velocity v. The mass reaches at its arrival at the starting point its initial velocity v. This velocity is the result of the work which the gravitational force mg, acting on m along the path v²/2g), has done; this work is

force·path = mg·v²/2g = mv²/2,

that is, it is exactly as large work as the mass can perform due to its velocity v (as has been shown above).

More about kinetic energy: If we lift with our hand the mass m with the weight mg dyn from the ground by h cm, we perform the work mgh erg. If we let m drop again from the height h cm to the ground, it arrives there with the velocity (2gh)1/2 ,whence it has the kinetic energy ½m·gh = mgh, that is, it can perform at this velocity the work mgh erg, work which is just as large as that which we have done to lift it. None of our work has been lost; the mass has returned it, although in a different form. Whether we can utilize this returned work or not is irrelevant. If we could do so, we would have received back our effort without a loss.

We now return to the equation p·h = mv²/2. Let p, p', p" ···be weights, m, m', m" ··· the corresponding masses, h, h', h" ···their depths of fall, v0, v0', v0" ··· their velocities attained. Then

ph = ½mv² .

If the initial velocities had not been 0, but v, v', v" ···, the summation would give the increase of kinetic energy due to the work performed

S ph = ½S m(v² - v0²).

This formula also applies when p is any constant force (not only a weight) and h is any path which has been traveled along (not only a fall height). If you know the entire path of the body during a motion and for each of its elements the force which performs the work, you can always employ this formula. But this knowledge is not always required: If the force is a central force - from a mass point at rest as centre and acting on another mass point with a magnitude which only depends on the mutual distance - then it is sufficient to know the distances of the starting and end points from the centre.

 You must interpret this as follows: If a body K is attracted to the centre C - according to any given law - you compute from this law the increase of the kinetic energy during a straight line approach to C from the initial distance r0 and the final distance r1. However, you obtain also the same increase when K moves in any way from the distance r0to the distance r1. Only the approach to C, the radial displacement, demands work, a tangential displacement (between points at equal distance from C) does not demand work. - The last equation above, applied to central forces, is called the Principle of Kinetic Energy.

This principle applies during application to purely mechanical processes, but only to frictionless processes; friction is not a central force! During processes involving friction, there arises less kinetic energy than would correspond to the work done. The part of the working force, which is lost during the mechanical process, appears as heat on the surface through friction - then the problem is no longer purely mechanical, it reaches beyond Mechanics and comes under the heading of Conservation of Energy, which embraces all physical processes. (Forces the work of which converts totally into mechanical energy or, in other words, the mechanical energy of which is conserved, are called conservative forces).

Principle of conservation of energy

In order to clarify the general relevant concepts, we will now consider quite a different problem side by side with that of lifting a weight and seek their common ground. In order to launch an arrow with a bow, you must first tighten the string. While doing so you work - the string becomes tighter the further you deflect it. The result of this work is the new shape of the string, the new position it has been given. When you release it, it flings back into its initial position and in doing so sets the arrow into motion; it works on the arrow, since it must overcome the arrow's inertia, in order to accelerate it.

We see: Just as you have returned the work, which the weight could perform while falling, to the weight by lifting it, you transfer initially the work, which the string can do by flipping back, to the string by deflecting it. Just as the falling weight receives its working capacity from its being previously lifted, the string, returning to its initial position, has taken it from the previously tightened string. In order to be able to do work, you had to place the weight first in a position of work capability. The expression being in a position to perform work corresponds to the physical process. You call the energy ( work capability) which a body has through its position its potential energy.

A mass falling from a given height can perform as much work as was required to lift it to that position.While tightening the string and letting it flip back we would reach the same conclusion, if we could trace them both equally well. The work, which you required while tightening the string, is completely returned to it, in that it relaxes and gives the arrow a velocity, which enables it to do as much work as required to give the arrow its velocity. Due to its speed, the arrow can overcome considerable resistance.

When this overcoming of resistance involves destruction, the performance becomes visible. However, if the work, required for the destruction, could be computed, it would not equal fully the working capacity of the arrow, which it has while encountering the resistance ( say, a wall); it is smaller. In other words, part of the work has been lost? No! A further examination would show: The impact of the arrow against the resistance has, besides destruction, provoked another action, a rise in temperature, that is, the arrow itself and its immediate neighbourhood have become warmer - it has done visible mechanical work as well as generated heat.

We will see later on that you can always generate a definite quantity of heat by a definite amount of work. One says: The generated heat is equivalent to that work, or that amount of performed (disappeared) mechanical work has changed into heat. Consideration of the equivalence of heat and work suggests that none of the work capacity of the arrow has been lost, but that the mechanical work of destruction plus the work used to generate heat equals the initial capacity for work of the arrow. The potential energy of the tightened string has vanished; instead, the kinetic energy of the arrow has arisen, which has converted itself into the kinetic energy of the scattered relics of that resistance and an amount of heat. None of the energy, placed into the system by tightening of the bow's string has been lost; whatever potential energy vanished, has reappeared as kinetic energy. It is only that the form of the energy has changed into another one - the total amount of energy has been conserved.

We have just seen how heat arose in plce of a certain amount of mechanical work, that is, instead of an amount of energy which certainly was not heat; we could even say: The work has changed into heat. That is why we call heat a form of energy. There exist other forms of energy: Light, magnetism, electricity, chemical forces which are closely related to mechanical energy. Throughout Nature, there exists a certain store of work, no matter whether it manifests itself by chemical or thermal or electric processes; it comprises partly kinetic, partly potential energy. If somewhere a quantity of potential energy disappears, it is replaced by an equivalent amount of kinetic energy. Arbitrary amounts of each of the two forms of energy can convert into the other, but not the least amount of either can disappear.

"An ... investigation of all ... known physical and chemical processes has shown, that all of Nature has a store of activatable force, which cannot be increased or decreased in any manner, so that the quantity of activatable force in inorganic Nature . . is eternal and unchangeable" (Helmholtz, who referred to energy by the term activatable force). This is the law of Conservation of Energy. The possibility of its general validity was first pronounced by the German medical doctor Mayer (Comments on the forces of lifeless Nature, Liebig's Annalen 1842). Its foundation forms the knowledge, gathered over hundreds of years, that it is in no way possible to construct a perpetuum mobile, that is, a machine, which without being wound up, without being driven by water, wind or other natural forces, can remain in motion eternally by generating internally its driving force.

The law of Conservation of Energy yields that exactly as much work, as a machine can perform, must be placed into it in one form or another, for example, as the energy of compressed steam, because a machine can only convert work and pass it on, but not generate it. However, not exactly that much energy is economically exploitable, because friction and other resistances convert a part of it into unreachable forms; yet this part is not at all destroyed, it only has become worthless for us. "Conservation of energy is the large, general rule which is in agreement with the facts not only of Physics, but of all science. Once comprehended, it becomes to the physicist the principle to which he can link all other known laws concerning physical effects; it places him in a position, to discover the lawful interrelations of such effects in new branches of his science. For these reasons this law is called in general the Principle of the conservation of Energy." (Maxwell, Matter and motion, Art. 73).

The Law of Conservation of Energy has also solved the question why a perpetuum mobile is impossible. In former times, it was believed that one had in the life of Man and animals a proof for it to be possible; however, Man and animals require food and food is a contribution of energy. Food is oxidizable, that is, it can be combusted1. A living being's body is therefore like a steam engine. Sugar, fats and starch perform in it what is done by ordinary burnable substances in the steam engine. Without their continuous supply, it cannot exist.

1. Only three chemical substances are oxidizable: 1. three simple sugars, 2. four fats, 3. seventeen amino acids, components of albumen.

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