B5. Motion on precribed trajectory

Stable, unstable, indifferent equilibrium

In order to discover how gravity affects the state of motion of a mass, you need only examine that of its centre of gravity. For this purpose, you imagine the mass to be concentrated at the C.G. For example, you may know that in spite of gravitational action a body remains at rest, if it is horizontally supported (or vertically suspended), because it receives vertically upwards the same propulsion which gravity gives it downwards. The body is in equilibrium. But this equilibrium can take different forms.

In Fig. 64, A is the highest, B the lowest point of a circular arc, C a point of the horizontal plane. At each of them, the mass can be at rest, because at A and B, the arc has the same direction as the horizontal tangent. If you displace the body only a very little bit - let the friction be minimal! - along the arc from A, it will slide along it and never return on its own accord to the original position of equilibrium. If you displace it from B, it also falls along the arc, but tries to return to the original position. Obviously, only a horizontal support and only such a support balances the action of gravity completely, while a body can rest on a curve only where the tangent is horizontal, that is, only at A and B; at all other points, the action of gravity is only partly balanced and the body follows the unbalanced part until it reaches the lowest point which it can reach. The equilibrium at A is called unstable¹.

^{1 This simple definition fails for moving bodies. A top which is not rotating and placed on its tip, is in unstable, one that dances on its tip is in stable equilibrium.}

^{If you suspend a sphere vertically like a pendulum and hold it at rest, it is in stable equilibrium; if you displace it from its equilibrium position and gravity acts on it, it tries to return to its equilibrium position, because its centre of gravity will then be in the lowest possible position. If it is balanced on a peak, it is in unstable equilibrium and falls with the slightest displacement. If it is like a globe, rotatable about an axis which passes through its centre, it is in indifferent equilibrium, because the centre of gravity is held fixed in space, that is, it cannot change its position; whatever its position, it stays at rest.}

^{The stability of a supported body (Fig. 65) can have different states.The equilibrium position of a beam can be I or II. In both, S denotes the centre of gravity. If I turned from I about its horizontal edge at A, it reaches at III an unstable equilibrium position. Similarly, when it is rotated from II about this axis. However, it reaches more easily from II than from I when turned about this axis. It will reach more easily from II than from I the unstable position, that is, II is less unstable than I. It will be more strenuous to move from I to II than from II to I.}

^{The stability is the greater the lower lies the C.G., that is, the higher it must be raised before the body reaches unstable equilibrium. A homogeneous, heavy, tri-axial ellipsoid is on a horizontal plane in stable equilibrium, if it rests on the end of its smallest axis, because every displacement raises its C.G; if it rests on the end point of the largest axis, it is unstable. A homogeneous sphere, a homogeneous circular cylinder on a horizontal plane are in neutral equilibrium.}

^{Equilibrium of a bifilarly suspended body}

The equilibrium of a body in bifilar suspension is also stable. If you suspend it in such a manner (Fig. 66) that the two threads lie in the same vertical plane and the straight lines between their upper ends (OO') and lower ends (UU') are parallel, the body is in equilibrium and at rest. If the first condition is not fulfilled, it oscillates like a bifilarly suspended swing; if the second condition is not met, it executes rotatotory oscillations. Its C.G. rest as deep as it can. If you rotate the body about the vertical through its C.G., this point rises the more the further you deflect the body out of its position of rest. When released, the body moves with a corresponding force back to its equilibrium position, passes through it, slows down, comes to rest, returns, etc., in other words, it swings about the vertical through its C.G. in the resting position, and finally comes again to rest in its original position of rest. Bifilar suspension thus gives a body an explicitly stable equilibrium. For example, it is employed when you want to let a magnetic needle rotate in the horizontal plane about a vertical line (magnetometer) or in the suspension of galvanometer coils, used in the electro-dynamometer of Wilhelm Weber. The distance between the threads depends on the desired sensitivity of the instrument. The returning force is generated in bifilar suspension by gravity , in unifilar ( rotating balance) suspension the torsional elasticity of the suspending thread.

The rotatable rigid body as machine

The roller

We return to the condition, under which two forces applied to a rotatable, rigid body balance each other. Such an arrangement is a machine. You can change the direction of a force by means of a disk which can turn about an axis at right angle to it; it is especially simple to do so with a circular disk the axis of which passes through its centre. The direction of the force is then always tangent to the circle and the distance from the axis of rotation equals the radius of the circle, the static moment of the force remains unchanged. A disk like in Fig. 67 is called a roller, in fact a fixed roller, because its axis cannot move. It is a wheel A which has a groove around its circumference. It can rotate about the axle e, which is through its centre and perpendicular to the plane of the wheel . The groove of the roller houses a string along which all forces act. In order to replace the pull of the hand by a weight P, you must make P as large as Q, because equilibrium will only occur when Pp = Qq and q = p = the radius of the wheel. Hence the fixed roller allows to change the direction of a force, but does not save effort. It is therefore not a machine by our earlier definition. But it is only not a machine, because all forces acting on it attack at the same distance from the axis.

Wheel on an axis

However, if you fix on a cylinder A which now becomes the roller, a second larger roller B (Fig. 68) in such a way that A and B can only rotate together, then the force Q acting on the smaller roller can balance a smaller force P acting on the larger roller. This is how functions a wheel on an axle. Pp = Qq yields P=(q/p)Q; the larger the distance p from the axis, compared with q, the smaller is P, which balances Q.

While the forces P and Q do not act in the same plane, you can transfer there (Fig. 59) every force parallel to it by addition of a corresponding pair of forces. The sense of rotation of the arising pair of forces is such that the axle DE tends to turn about an axis perpendicular to it so that it is neutralized as in Fig. 60 by the bearings D and E of the axle.

The block-and-pulley

Rollers as in Fig.54 or in Fig.67 can only alter the direction of a force, but they can act as machines in the sense of the definition, if they can move altogether (free roller, Fig. 69). Imagine it to be held by a string in its groove, so that both sections of the string are parallel and a force Q acts vertically downwards on its axle, say a weight Q. Let the roller itself be a weightless, geometric structure. We can replace the force Q by two forces Q/2, which act on the rim of the roller. On each half of the string, the weight pulls downwards with the force Q/2. In order to maintain the roller in equilibrium, each hand must pull its string upwards with the force Q/2. If you fix one end of the string as in Fig. 69, the other hand must only pull upwards with the force Q/2, that is, applied to the circumference of the roller, the force Q/2 is sufficient for holding the load Q in equilibrium. If you lead the free end over a fixed roller you obtain a block-and-pulley (Fig. 70). If you guide it to the axle of a second, free roller, which like the first one is supported by a rope, then you can balance a load Q/4 by fixing one end of the rope. You need only apply to the free end of the rope Q/4 = Q/2². If you guide the free end of the rope to a third roller, then Q/8 = Q/2³ is required to hold it in equilibrium.With n movable rollers, you need only apply to the nth roller the force Q/2ⁿ.

For example, if Q = 200 kg, you only need apply 100 kg to the block-and-pulley in order to establish equilibrium; you can then readily make it move: The mass of 100 kg sinks, the one of 200 kg rises. However, you must then lower the 100 kg twice as far as the 200 kg will rise. 100 mkg = 200/2 mkg! In other words, you do not save effort, the block-and-pulley only lightens the work, but in the same way that it eases it, it lengthens it as every other machine.

All block-and-pulleys employ fixed and free rollers. In the common block-and- pulley (Fig. 71), the rollers are combined in threes in a bottle. When a block-and- pulley is in equilibrium, each roller must be at rest, whence, apart from the rope not being exactly parallel, the forces pulling on the two sides of a roller must be equal, that is, Q₁ = Q₂, Q₂ = Q₃, Q_{3 =} Q₄, that is, the pull which Q exerts downwards is uniformly distributed to all sections of the rope and since there are as many sections of rope as rollers, the force P which balances Q is Q/n. The differential block-and-pulley (Fig. 72) is used most frequently. The fixed roller consists of two differently sized rollers; their different radii determine the ratio P/Q .

The lever

If you reduce the disk (Fig. 67) to the bar SCT, you create a straight lever in contrast to the angle lever SCT (Fig. 74). The distance of the point of attack (S,T) from the axis is called the arm of the lever in the case of the straight lever SC and TC, in that of the angle lever S'C and T'C - it is called a two-arms lever, if C, as in Figs 73 and 74, lies between the points of application S and T, otherwise a single-arm lever (Fig. 75).The equilibrium condition is the same as for every rigid body which can rotate about a fixed axis: The forces must rotate in opposite directions and their static moment must be equal, that is, the forces P and Q must satisfy the equation Pp = Qq. In contrast to a mathematical lever, the bar ST is called a physical lever. If you imagine ST to be replaced by a rigid, weightless straight line, you arrive at the concept of the mathematical lever. (Its introduction in place of the physical one brings certain advantages. A physical lever is subject to gravity! In the sequel, the physical lever may occasionally be replaced by a mathematical one without special mention.)

The lever as a tool

We could have equally well used as a lever instead of the disk a bar or a body of any other shape. Fundamentally, provided it has for the purpose a suitable shape, every body which can be rotated about an axis can serve as a lever. To each applies the equation Pp=Qq. In any given case, you must ask: Where lies the axis of rotation, at what points act P and Q, in what direction do they rotate the lever? In Fig. 76, the lever is a straight, inflexible bar which serves to overcome the weight of the block and lift it. In order to balance the force Q, P must act in the direction of the arrow. The axis of rotation, the length of the lever and its sense of rotation are clear. The longer the lever bar, the smaller you can make P. - The barrow in Fig. 77 with the load Q is a one-arm lever. The axis of rotation is the axle of the wheel. The longer the handles and the closer lies Q to the wheel, that is, the smaller is q, the less force is required to raise the barrow to the position required for pushing it along. - Many of our tools are levers, frequently combinations of several levers. Pliers consist of two levers (Fig. 78) You only need to imagine that the disk of Fig. 73 has been reduced instead of to an angle or a bar to one of the halves of the pliers, in order to understand the action of pliers by combination of two levers. The arrows indicate the directions of the forces exerted by the hand compressing the pliers. - Most neighbouring bones in our skeleton have joints which convert them into levers. Their motions relative to each other, for example, those of the lower arm relative to the upper arm (Fig. 79), when it moves in the elbow, joint like a door hinge, are conditioned by the form of the joints: They are rotations. The axis of rotation is the axis of the joint, while the bones, which can rotate about the axis of the joint, are levers. The motions of the bones are caused by the muscles. These are stretched between two independent points, in Fig. 79 between A and P. As they contract, they turn the lower arm towards the upper arm. It is an one-arm lever; the points of attack of the load W and the force P lie on the same side of the axis through F.

The lever as a balance (measurement of mass)

The beam balance employs a lever. It gives you the magnitude of a mass (not its weight !), that is, how many gram it contains and compares for this purpose the mass to be weighed M_x with a known mass M. You select the mass M out of a set of masses of known magnitude, for example, 50, 20, 5, 2, 1 g. You compare the unknown mass M_x with the known mass M by (Fig. 80) placing it at one end and the known mass at the other end of a two-arm lever which can rotate about a horizontal axis. At one end of the lever the weight of the mass M_x pulls downwards, that is the force M_x·g, at the other end the force M ·g. If M_x is at the distance l_x from the axis and M at the distance l, the lever is in equilibrium when M_xg·l_x = Mg·l. If M_x and M have the same distance to the axis, that is, l_x = l, the beam of the balance is equal-armed and the equilibrium condition is M_x=M. The unknown mass then equals the known mass. - If l_x is not equal to l, that is, the masses are at different distances from the axis, (Fig. 81), then the equilibrium condition is M_x·l_x = M·l. For example, with l = 10 l_x, when the known mass M is ten times as far away from the axis as the unknown mass, one has M_x = 10·M, that is you can weigh a large mass by means of a comparatively small mass (cf. Bridge balance).

The design of balances depends on the desired accuracy and magnitude of the weight to be measured. The principle of lever balances - equality of static moments - does not change.

A typical precision balance is shown in Fig. 82. In order to be reliable, it must meet several conditions.
1. Without load, its lever must be in equilibrium, that is, the pointer Z must be at 0. The arms of the lever and the scales P and Q must be as equal as possible, in order to generate equal static moments about the axis of rotation. By turning the screws K and L, that is, by changing their distances from the axis of rotation, you can correct the moments of the two sides.
2. Interchange not too small (This weights which balance the scale from one side to the other without changing the location of Z indicates that the two to the arms are equally long).
3.The lever must be in stable equilibrium, that is, it must return on its own equilibrium position, if the it is deflected and then released. Hence the centre of gravity of the lever must lie vertically below axis. (If it were to lie above it, it would leave the equilibrium position at the slightest deflection.) If it were located on the axis, then the beam, as both arms are equally loaded, would be in equilibrium in every position, if differently loaded, it would flip completely to the side with the larger load. However, its stability should not be larger than necessary, since it would lower the balance's sensitivity, whence the centre of gravity should be just below the axis of rotation.
4. A sensitive balance should deflect strongly for a slight difference in the two weights. The sensitivity is measured by the angle by which it deflects, if it is on both sides equally loaded, for example, when one side has 1 mg more than the other; it decreases with increasing load except when the conditions in Fig. 83 are fulfilled. The sensitivity - we do not give a proof here - is the larger, the longer the arms of the lever, the smaller their weight and the closer the centre of gravity lies to the axis of rotation. (The nut N allows you to move the centre of gravity closer to the axis of rotation. The formula for the sensitivity is:

C=(L/(Ma+M'a')·(z/a),

where M is the mass of the beam, M' the masses of the scales and their loads, a the distance of the axis of rotation from the centre of gravity, a' the distance of the axis of rotation from the plane (Fig. 83) with the knife edges, z the length of the pointer, s the length of the subdivisions of the scale.
5. If the sensitivity is to be the same for all loads, this is only possible when (Fig. 83) the bearing edges of the three knife edges bac lie in the same plane. Only then will the resultant of the two forces, acting at b and c, pass also at deflection of the beam through the fixed bearing edge a and will not affect the displacement of the beam.

Beside the sensitivity, the symmetry of the two sides and the complete stiffness of the beam of an equally armed balance is most important. In order to eliminate any asymmetry of the beam arms and scales, you use the method of double weighing ( named after Borda 1733-1799): The body to be examined is placed on one scale and sand or a similar substance on the other scale until equilibrium has been achieved. The body is then replaced by known weights. Thus, the target mass is compared with weights on the same arm of the lever and in the same scale, that is, independently of symmetry. - The need for perfect stiffness of the beam, firstly, limits its length and, secondly, makes it not too light. Therefore Bunge's precision balance improves the ratio length/mass of the beam and thus the sensitivity of the balance ( 4. ) by reduction of the length and a special form of beam.

In the portable balance (Fig. 81), you move the weight M until equilibrium is reached with the load hanging from the short arm. In the automatic letter balance, in which you induce equilibrium by changing the inclination of the beam, the weight S rises as a load is imposed and increases its distance from the vertical passing through its axis until the static moment equals that of the load. You read the result on a scale. - The micro-balance of Walther Hermann Nernst 1864-1941 also employs inclination to achieve equilibrium. The beam is a hooked quartz thread, pasted on to a quartz thread, vertical to the plane of the hook as central knife edge. It yields weights as small as 10^-6 mg.

Bridge balance (H.Quintenz, 1822)

Large weights are measured on bridge balances (Fig. 85). In order to weigh correctly, it must meet two conditions:
1. The result must not depend on the position on the bridge, where the body is placed;
2. The weight on the scale must be in the prescribed ratio to the body to be weighed (1:10 for the decimal balance). The balance comprises three levers, the two-arms ACD and two one-arm ones HK and EF. The two-arm one is the balance beam: It carries at A the scale for the weights, at B (with the aid of the bar BH) the bridge HK with the body to be weighed. The bridge HJ is one of the one-arm levers with its axis of rotation K. It is not totally fixed. The load L is supported at the points H and K and distributed between them in any manner which you need not know. Let p and q denote these forces (at A a pull and at K a pressure), so that L=p+q. You can imagine the weight q at K, which acts at the lever KF, to be replaced by a weight q/n at E, if you let EF = n·KF. It acts at E at the lever arm CD of the two-arms lever. If you make the lever arm CD=n·CB, you can replace the weight q/n at E by n times as large a weight q at B. - then both p and q act on the same lever arm CB. The load on the bridge acts therefore independently of its position on the bridge as if it were hanging from B. ( For the weighing not to depend on the position of the load on the bridge, you must also have that DC = n·BC, if EF = n·KF, that is, EF : KF = DC : BC). - If you now set AC = 10·BC, the second condition is also met, namely that the weight is 10 times as far from the axis of rotation as the load, whence the ratio weight/load at equilibrium is 1:10.

Lever balances can only determine the mass of a body (gram), not the weight (dyn). The condition of equilibrium of lever balances was M_xg·l_x = Mg·l. But at the Equator, g is smaller than at a Pole, whence the same body is lighter at the Equator than at the pole. A mass with an exact weight of 1 kg at Latitude 45º, weighs at a Pole 2.6 g more and at the Equator 2.6 g less. The lever balances do not show this, because a change of g changes both sides of the balance in the same way: M_xg·l_x as well as Mg·l, that is, the rotating forces on both sides of a lever balance are the same everywhere on Earth, if they are at one point. Since a lever balance cannot measure the change of a weight, it also cannot measure the weight itself. In order to measure the weight of a body, that is, the number of force units (dyn) by which Earth attracts it, you can employ a spring balance

Lever balance as dynamometer (Prony's Brake)

The dynamometer of Gaspard Clair François Marie Riche de Prony 1755-1839, by means of which engineers measure the performance of machines, also employs the principle of a lever balance. You make the machine overcome in its clamp the single work of a resistance and ensure that it works with the prescribed number of turns per friction second (that is, you replace the machine's useful work by the calculable one for overcoming a measurable friction. The Brake must therefore 1. load the machine, 2. measure the load. These two functions do not depend on each other. - The clamp (Fig. 86) comprises two symmetric clamps M and M' which are used to load the machine. For this purpose, you tighten the screws V until you reach the intended number of revolutions. The axle then rotates with strong friction between the clamps. The energy which is required for this rotation is enough to overcome the friction. (If you were to interrupt the supply of energy, the machine would stop immediately.) The lever arm with the weight scale and the weights P serves the purpose of measuring the static moment of the rotating axle. For this purpose, you load the weight scale until the lever is in equilibrium. How does the equilibrium arise? Friction acts at each point where the axle touches the clamps. The frictional forces around the clamp try to turn the clamp in the direction of rotation of the axle . All of them act in the same direction. You can therefore sum them and imagine them replaced by the sum of the forces which act on the upper clamp; Let F denote these forces and r is the radius of the axle, then Fr is the static moment which acts on the upper clamp; if of of the force, which tends to turn the clamps. The static moment in the opposite direction of the weights P in the scale plus the weight p of the clamp is PL + pd (where d is the distance of the centre of gravity of the brake from the axle). Since we have established equilibrium, Fr = PL + pd. If v is the rate of rotation , a point of the surface of the axle covers in one second the distance rv. The complete circumference of the axle overcomes the frictional force F, the load along the distance rv and thus performs the work A=F·rv = (PL + pd) mkg*/sec, where the distance is measured in m, the load in lg. If n is the number of revolutions per minute, then v = 2pn/60, that is, A = (2pn/6)(PL + pd) mkg* or (2pn/60·75)·(PL + pd)HP . You need not know the magnitude P of the friction between the axle and the clamps as it cancels during the calculation. The Prony Brake is a dynamometer (instrument to measure force) and due to its mode of action is a brake-dynamometer.

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