The vertical line through the centre of gravity of the polygon must clearly just pass through the lower point of the side which is in contact with the plane, but the centre of gravity of the polygon is manifestly the centre of the figure; therefore in this polygon the line joining the centre of the figure with one of the angles, must make with the adjacent side an angle which is the complement of 10°. But this angle in a regular polygon must always be the complement of the 180°

angle which each side subtends at the centre, i. e. of n if n be the number of sides of the polygon. Hence we have here

or the number of sides required is 18.

(31). Why does a person carrying a heavy weight in his hand lean towards the opposite side? (Art. 37.)

(32). When a man rises from a chair, why does he bend his body forward and his legs back?

(33). If a sphere were not homogeneous (not made of uniform material) how could it be practically ascertained whether or not its centre of gravity were at the centre of the figure? (Art. 37.)

SECTION V.

FRICTION.

39. WHEN two surfaces, not smooth, are in contact, and it is attempted to make the one move upon the other, a force due to the want of smoothness arises and tends to prevent the motion; this force is Friction. It acts upon each surface in the tangent plane common to both surfaces at the point of contact, and in a direction exactly opposite to that in which the other forces tend to make this point move.

In general, just so much of this force is called into action as will serve to keep the point of contact at rest, but there is a certain limiting value in each case, beyond which it is found that friction cannot be exerted: this limiting value is always proportional to the normal reaction at the point of contact, the proportion depending only upon the nature of the two surfaces.

Thus, if P be the normal force at the point of contact of two surfaces in a supposed case, F the greatest amount of friction that can be exerted at that point with the normal force P, then F= μP where μ is a numerical quantity which is constant, so long as the materials of which the two surfaces are formed are the same. This quantity μ is termed the coefficient of friction of the particular surface to which it refers, and its value can only be obtained by experiment.

If the surfaces be in contact throughout a plane area, the relations just mentioned will hold between the limiting amount of friction and the normal pressure at each point of it: and hence it can be easily seen that the same must be true for the resultant of the pressures and the resultant of the limiting frictions; the coefficient of friction thus remaining unchanged is independent of the extent of surface in contact.

40. Suppose A to be the point of a given surface which is in contact with another, and let AB, AC represent the normal force P and the friction Frespectively acting upon A; they are therefore in the directions of the normal, and tangent to the surfaces at A.

B

Р

ε

D

Draw AD the diagonal of the parallelogram described upon AB, AC; then AD represents both in magnitude and direction the resultant of P and F. It is clear that the larger F is for the same value of P, the farther will AD lie from the normal AB, and that its greatest angular distance from AB will correspond to the greatest value of F, i.e. to FμP: let AC' represent this value of F, and AD' the resultant in this case of P and F; then

A

The mag

It is usual to represent this angle by the symbol ɛ. nitude of the resultant evidently equals √(P2 + F3), and may therefore be as great as the circumstances of the case require, for P can always be exerted to any extent.

41. It thus appears that when two surfaces are in contact, their resultant reaction upon each other may be anything whatever as to magnitude, and also as to direction within a certain limiting angle with the normal, which we have called ɛ. This is often by far the most convenient light in which to view the effect of friction. The best way of finding μ for different substances is to observe this angle ; there are many devices for effecting this object.

42. Our definition of perfect smoothness (Art. 9, 8) is equivalent to μ = 0, and therefore e equal 0. μ put equal to xc, and

π

therefore E = gives a case where the one surface could not , 2

possibly slide upon the other; the surfaces are then termed perfectly rough. Both these may be looked upon as limiting cases of the action of surfaces upon one another, for they neither of them actually occur in nature, although we often find extremely near approximations to them.

43. When a body is in contact with a plane surface, the contact existing at any number of points in the same straight line, and neither surface being supposed to be perfectly smooth, since the reactions at the several points will not be necessarily perpendicular to the plane, their resultant will depend for its direction as well as its magnitude upon the circumstances of the particular case; it may have any inclination to the normal to the plane within a given limit (the ɛ of Art. 40) and be of any magnitude, but it must still pass through some point in the line in which lie the points of contact, and between the extreme points of contact. Hence equilibrium will be always preserved if the resultant of the other forces acting upon the body, whatever be its magnitude, pass through some point in this finite line, and have any direction within the abovenamed limiting angle ε.

44. In the supposed case of perfect roughness, the reaction at each point of contact only differs from that which can be exerted upon a fixed point, in that it cannot be towards the plane, in other respects it is indeterminate; hence it is only necessary for equilibrium, that the resultant of the other forces acting upon the body should not act from the plane, and should pass through some point in the before-mentioned line.

A full explanation of friction, with methods for determining its amount, and tables giving the values of μ and for a great number of different substances, may be found in the Third Treatise on Mechanics, published by the Society for the Diffusion of Useful Knowledge.

Examples to Section V.

(1). A particle of weight W rests upon an inclined plane whose coefficient of friction and inclination are given; the particle is attached by an extensible string to the top of the plane; the tension of the string is always in a constant proportion to its length: find the greatest distance from the top of the plane at which the particle will remain in equilibrium.

Let AB be the plane inclined at an angle a to the horizon, tan & its coefficient of friction, x the distance of the

particle from B when the greatest amount of friction is exerted, i.e. in the supposed case; F the tension of the string at this length, R the reaction of plane, making angle with the normal at C.

The particle C is kept in equilibrium by the forces F, R, and W. In CW take any point K, and draw KM parallel to CR and meeting AB in M.

Then the sides of the triangle CKM are parallel to and therefore proportional to the three forces which keep C at rest; we have therefore