We see therefore that, for the balance to be true, the arms must be of equal length (a=b) and the scale-pans of equal weight (S1=S2).

In order to obtain the conditions for sensitiveness we may suppose that the load in the right pan is P and that in the left P+x, and that the beam takes up the position A'BC', making an angle ✪ with the horizontal. The turning moments in the positive direction is then equal to x.DB, while that in the negative direction is w.HG', where w is the weight of the beam and HG' is the distance between its centre of gravity in the displaced position and the vertical through the central knife-edge. Since the angle GBG' is equal to 8, we have

HG'=BG' sin 0 = sin 0,

where is the distance of the centre of gravity of the beam below the central knife-edge.

Now the greater the value of tan 0, the greater must be the angle 0, i.e. the greater the deflection. Thus for a given difference (a) in the loads of the pans the magnitude of the deflection will depend on the magnitude of the fraction This fraction is increased in value if we

a

wh increase a or decrease w or h. Therefore, in order that the balance may be as sensitive as possible, so that a readable deflection may be produced by a small difference in the masses placed in the pans, we must make a, that is the length of the beam, as large as possible, and make w, the weight of the beam, and h, the distance of the centre of gravity below the central knife-edge, as small as possible.

In order that a balance may be quick in returning to its position of equilibrium after being displaced, it is necessary that, when displaced, the moment tending to bring the beam back to its equilibrium position should be as large as possible. Since, when the pans are equally loaded, the only turning moment is that due to the weight of the beam, to secure quickness of vibration we must make the quantity wh sin 0, which expresses this moment, as large as possible for every value of 0. We can do this by making h large. No advantage would accrue by making w large, since, although we should thereby increase the turning moment, we should increase the mass to be moved in the same proportion, so that the acceleration with which the beam would move, and hence the time taken to return to its equilibrium position, would remain unaltered.1 The 1 The reason for this will be more apparent when the subject of the time of vibration of a pendulum has been discussed. See § 113.

only way, therefore, of securing rapidity in the indication of a balance is to make the centre of gravity of the beam some distance below the central knife-edge. It will be observed that this condition is in direct antagonism to one of the conditions for sensitiveness, and we have in this case to choose such a value for has will make the balance fairly quick, without unduly reducing the sensitiveness. Another element which affects the quickness of a balance lies in the fact that when the beam moves the mass moved includes not only the beam itself, but also the scale-pans and their contents. It will be evident that for a given load the distance through which the load is moved, as the balance beam swings, is greater if the beam is long. Hence lengthening the beam will increase the time the balance takes to swing. This requisite, again, clashes with one of the requisites for sensitiveness.

The easiest way of drawing attention to the way in which the different requisites of a good balance are secured in a modern physical balance is to describe such a balance; and of the many slightly different types in use we will select one of those made by Bunge. The beam of the balance consists of a triangular girder-shaped framework ABC (Fig. 70). This framework carries the central knife-edge H and the end knife-edges A and B. It also carries in front a notched cross-bar DE, on which the rider can be placed, and a long pointer F. An upright rod attached to the back of the beam serves to counterbalance the pointer, &c., in front, and carries two small weights I, by means of which the position of the centre of gravity of the beam may be raised or lowered, and hence the sensitiveness altered; by moving the weight on the horizontal arm, the beam can be made to balance in the horizontal position when there is no load on the pans. The stirrups which carry the pans have small agate planes, which rest on the terminal knife-edges of the beam. The stirrups also carry two small agate points which, when the beam is lowered, fall, one into a small conical hole, and the other into a groove, which are carried by uprights K attached to the stand. These serve to slightly raise the agate planes from the knife-edges when the beam is lowered, and thus prevent the knife-edges being damaged when weights are being placed on or removed from the pans. The beam itself, when lowered, is raised from the central plane by two similar agate pins, as well as by two knife-edges LL, which support the arms. The position of the beam is read by the pointer F, which moves over an ivory scale. For very accurate work, where the smallest movement of the pointer has to be observed, a microscope M is employed, which is focussed on a small, finely divided scale G attached to the pointer. The handle N serves to raise and lower the beam, and to raise the supports which come up and catch the lower surface of the pans when the beam is lowered. A small lever O, worked by the handle P, serves to adjust the position of the rider. The rider itself weighs only half a centigram, and the position of the adjusting weights, I, is so chosen that the beam is horizontal when the rider is at

the extreme left-hand end of the beam, and no weights are in the pans. Thus when the rider is at the centre of the beam it is equivalent to a weight of half a centigram in the right-hand pan, while when it is at the extreme right-hand end of the beam it is equivalent to a centigram in this pan. The object of this arrangement, rather than the more usual one where the rider weighs a centigram and only moves over half the length of the beam, is that the scale along which the rider moves is twice as open. This is of importance, since the length of the beam is

only thirteen centimetres, so that otherwise the movement of the rider, corresponding say to a tenth of a milligram, would be so small as to be hardly observable. The advantage of the short beam is that the time the balance takes to make a swing is much smaller than would be the case with a long beam, so that the time taken to make a weighing is thereby much reduced. By the employment of a very long pointer and the microscope, we make up for the sensitiveness lost by the use of a short beam.

影

CHAPTER XII

FRICTION

96. Statical Friction.-Suppose that a body C (Fig. 71), of mass m, rest upon a horizontal plane AB. Then, if no force except gravity acts, C will be in equilibrium under the action of two forces-(1) the weight mg of the body acting vertically downwards, and (2) the reaction of the plane, which must act vertically upwards and be equal to mg. Now, let a force P act on C, parallel to the surface AB. It is found that unless P exceeds a certain value the body still remains at rest. Under these circumstances the body is in equilibrium under its weight mg, the force A

This

R

P

Fi

C

B

W

FIG. 71.

P and the reaction between its surface and the plane, which must now be inclined to the normal, and act in some such direction as CR. force along CR' may be resolved into a reaction normal to the surface, i.e. along CR, and a force along CF which must, if there is equilibrium, be equal in magnitude to P. This force, which is brought into play when we attempt to slide one body over another, and which always acts so as to resist motion, is called the friction between the surfaces.

If the total normal pressure between C and the plane be Q, then it is found that C will commence to slide when the force P bears to Q a certain ratio, which is necessarily less than unity. This ratio is called the coefficient of friction between the body C and the plane AB, and is generally denoted by the symbol μ. The value of the coefficient of friction is independent of the size of the surface of contact between C and AB, and of the pressure Q. It depends, however, on the nature of the substances forming the two surfaces in contact, on the smoothness of these surfaces, and on the presence or absence of any lubricant, such as oil, fat, blacklead, &c., between the surfaces. The value of μ has to be determined experimentally for each of these conditions. If the force P is less than μQ, then there will be no frictional resistance F will be equal and opposite to P.

motion, and the When P is just

equal to μQ motion will be on the point of taking place, and the frictional

resistance will have its maximum value (μQ). If P is greater than μQ motion will take place, but the moving force will be less than P, since, although when motion has commenced the frictional resistance is often no longer equal to μQ, yet friction still acts as a force tending to prevent motion.

Since the coefficient of friction is independent of the surface of contact, it follows that for a given value of Q the frictional resistance (F) is also independent of the extent of the surface of contact. If A is the area of this surface, then the pressure per unit area is Q/A, and the frictional resistance per unit area is μQ A. If, while Q remains the same, a is reduced to A', then the pressure per unit area is increased to Q/A', and the frictional resistance per unit area is increased to μQ/A'. Hence the frictional resistance per unit area varies directly as the pressure per unit area.

97*. Limiting Angle.—When motion is just about to commence, and hence P is equal to μQ, the body is in a state of equilibrium under three

forces, the force P acting horizontally, the pressure Q acting vertically downwards, and the reaction acting along CR' (Fig. 72). In order to find the angle which CR' makes with the normal, we draw a line DE parallel to Q, and of such a length that it represents FQ in magnitude, and

from E draw EF parallel to P, and hence at right Then, by the triangle of

angles to DE, to represent P in magnitude. forces (72), the reaction which, together with the forces P and Q, maintains the body c in equilibrium, must be represented in magnitude and direction by the line FD. Therefore the angle FDE is equal to the angle between the reaction CR' and the normal. Since DE is equal to Q and EF to P, which is equal to μQ, we have

This angle 4, which represents the greatest angle the line of action of the reaction can make with the normal to the surface of contact, is called the limiting angle.