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 h as 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 1, 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,

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

R

R

P

F

C

B

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 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'. This 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.

W

FIG. 71.

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 Since DE is equal to

between the reaction CR' and the normal.

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.

F

N

If a force is applied to C along such a direction as FC (Fig. 73), making an angle of with the normal, then if is less than the limiting angle, motion of C will not take place, however great the value of this force. The reason is that we may resolve the force into two components, one parallel to the surface, which tends to produce motion and is resisted by the friction, and the other, which acts along the normal, produces a contact pressure. If F is the force, the component parallel to the surface

A

FIG. 73.

B

is F sin, and the component parallel to the normal is F cosy. If motion is just about to take place, and we neglect the weight of the body, then

F sin μF cosy,
μ=tany.

But μtan

where & is the limiting angle.

Hence if is less than $

motion will not take place.

98*. Angle of Repose.-If a body G (Fig. 74) of mass m is placed on an inclined plane AB, then, if there were no friction between G and the plane, the only forces acting would be the

weight, which is a force of mg acting verti

cally downwards and the reaction of the plane

R

C

D

E

FIG. 74.

GR acting at right angles to AB. As these forces are not in the same straight line, the body would move down the incline. If, howAever, there is friction between G and the surface of the plane, the friction will tend to prevent motion, and till the plane has a certain slope the body will remain at rest. To find the maximum inclination (4) of the plane to the horizontal we resolve the force mg into a component parallel to BA, which tends to produce motion, and a component normal to BA, which acts as the contact pressure. In the triangle DGE, the angle EGD is equal to 4, and ED is parallel to AB. Hence the component of mg parallel to BA is mg sin, and the component perpendicular to BA is mg cos y. If motion is just about to commence,

mg sin umg cos
μ=tan .

Hence if is greater than the limiting angle, motion takes place. The maximum inclination to the horizontal of the plane which is possible without the body sliding is called the angle of repose.

Thus the angle of repose is equal to the limiting angle, and the coefficient of friction is equal to the tangent of either of these angles.

H