(12) A triangular tripod has its three legs mutually at right angles; if their lengths be 15, 10, 12 ft. respectively, find the distance of the apex of the tripod from the ground. (If a, ß, y be the direction angles of the perpendicular p from the apex to the ground, then 15 cos a=p &c.)

SECTION (III). APPLICATIONS TO CONCURRENT FORCES.

55. Even at this early stage in our Theory of Vectors we can give some account of its application to concurrent forces. Those who are well grounded in Elementary Statics may omit this section and proceed at once to the chapter on Mass-centres.

If a number of forces act at a point, then we know from experience that a single force can always be found which acting at that point would produce equilibrium. This force is called the Equilibrant of the given forces. It is also a matter of experience that two equal and opposite forces acting at a point are in equilibrium. Hence a force equal and opposite to the equilibrant of a given set of forces would produce the same effect (as far as motion is concerned) as the given set of forces, this force is called the Resultant of the given forces; and in relation to this resultant the given forces are called the Components.

56. The Resultant of a number of forces acting at a point passes through that point and is given in magnitude, direction and sense by the sum of the vectors representing the forces.

This may be experimentally verified as follows.

Three strings are knotted together at one point, two of them pass

1

over smooth pulleys A and B and support weights W1 and W2, the third string hangs vertically and carries the weight W.

If the sum of any two of the weights be greater than the third, the system will take up some position of rest AOB as in the figure.

If the pulley be fixed to a vertical drawing-board covered with paper, the positions of the strings may be marked off.

Draw to any convenient scale O1B1 parallel to OB

and proportional to W2, then BA, parallel to OA and proportional to W. Then it will be found that A11 is vertical and proportional to W.

The forces represented by these lines are in equilibrium and the vectors representing the forces form a closed vector polygon.

Hence if three concurrent forces are in equilibrium, their vector polygon is closed.

The converse is also true, viz., if the vector polygon be closed, the three forces are in equilibrium.

This is known as the triangle of forces.

If we draw any triangle ОB11 and to the three strings attach weights (as in fig. 34) proportional to the lengths of the sides, the system of forces will be in equilibrium.

then

and

Let a denote the force in OA, ẞß that in OB, and y that in OC,

a+B+y=0,
a+B= −y.

Hence -y is the resultant of a and B, and we see that the resultant of two forces in magnitude, direction and sense is obtained by adding the vectors representing the forces, the position of the resultant being determined by the point at which the forces act.

[It is not necessary that one of the forces should be vertically downwards, by passing the string OC over a third pulley its direction could be altered at pleasure.]

57. Since we can always combine any two forces as explained above, the vector-polygon theorem may be extended from three to number of concurrent forces.

any

If a number of concurrent forces are in equilibrium, their vectorpolygon is closed.

If the vector-polygon be closed, then the forces acting at a point are in equilibrium.

58. We will now consider some examples.

(a) A weight W is placed on a smooth inclined plane and is kept at rest by a force acting up the plane. Determine the magnitude of this force and the pressure on the plane.

By a smooth surface is meant one that offers no resistance to bodies sliding on it, that is, there is no force along the surface. Any force there is between two smooth bodies must therefore be at right angles to the common surface.

The action between the body and the plane must be a compressive strain, the plane pushes against the body with a force R, say, and the body presses on the plane with a force - R, and since the plane

is smooth, the direction of these forces must be perpendicular to the plane.

We may suppose for clearness that the force acting up the plane is applied by means of a string carried over a pulley A and supporting a weight P.

Since there is equilibrium, the vector-polygon of W vertically downwards, P along the plane, and R' to plane must be closed.

Represent W by the vector y, the length of y being some convenient number of centimetres or inches. Suppose W is 10 lbs., then y may be drawn 2 inches long, so that 1 inch represents 5 lbs. Draw through the two ends of y lines parallel and to the plane, and so obtain the vectors a and B, where

a+B+y=0.

Hence a represents the tension in the cord and ẞ the reaction R of the plane, both in magnitude and direction. By measuring a and ẞ in inches and multiplying by 5, P and R can be found in lbs.

Of course, the magnitudes of a and ẞ could in this case be easily calculated when the inclination of the plane is given. By measurement P=4.8 lbs. approximately and R=6·6 lbs. approximately.

(b) A weight of 10 lbs. is suspended from a hook by a string 3 ft. long, a force equal to a weight of 6 lbs. acts on the suspended body horizontally, find the tension in the string and the position of equilibrium.

In this case two forces are known completely, the third is unknown in magnitude and in direction.

The process is as before; take say 1 cm. to represent 2 lbs., and set off 3 cm. horizontally, 10 vertically, then on measurement the third side of the vector-polygon will be found to represent 11.67 lbs. Let A be the point of suspension, then through A draw a line γ and this will give the direction of the suspending cord.

parallel to

If AP represents to some scale 3 ft., then PM will be found to

represent 1.54 ft. approximately, the horizontal displacement.

(c) A string 6 ft. long is fastened to two points at the same level, the distance apart of the points being 4 ft. A weight of 7 lbs. is suspended by a string knotted to the first string at a point distant 2 ft. from one end. Required the tensions in the two portions of the first string.

The form of the string AOB is found by the intersection of two circles of radii 2 and 4 ft. respectively.

Set off y to scale parallel to OW and through the end point draw a and ẞ parallel to OB and 04 respectively; by measuring a and B we find that the tension in OB is about 1.8 lbs. and that in OA is 6.2 lbs. nearly.

If the weight had been suspended from the string AOB by means of a smooth ring, then the tensions in AO and BO would have been equal.

59. Friction. However carefully the experiments mentioned in § 56 are carried out, the results obtained will never be quite in accordance with the law given. The more carefully the experiment is performed, the smoother the pulley or the better the bearings are lubricated, then the more nearly will the experiment agree with the rule. The discrepancy arises from the fact that the smoothest surface obtainable does offer some slight resistance to sliding motion.

Always when one body slides or tends to slide on another the motion is resisted; this resisting force is called the Friction between the surfaces.

60. The laws governing the friction between bodies at rest may be experimentally determined as follows.

AB is a horizontal board whose upper surface is made of the substance whose friction with other surfaces is to be determined. C is a slider whose lower surface may be changed. On C may be

placed weights of any desired magnitude, to C is attached a cord which passing round the pulley D supports a scale-pan P. If the bearings of the pulley D are well lubricated we may neglect the friction there in comparison with that between C and AB.

It is found that P can be loaded up to a certain extent without any motion taking place. During this loading up of P the friction between C and AB must have been increasing, for since there is equilibrium the force of friction must just balance the total load in the pan. The biggest load that can be put on the string without motion taking place measures the maximum amount of friction that can be called into play between the surfaces and is generally called the limiting friction.

By altering the surfaces in contact, the weight W, and the size of the slider, it has been found that the limiting friction is

(i)

dependent on the nature of the surfaces,

(ii) independent of the area of contact,

(iii) proportional to the weight W + that of the slider, i.e. to the normal pressure between the surfaces.

These laws hold between fairly wide limits.

If R denote the normal pressure and F the limiting friction, then from (iii)

and

μ

F=μR,

is called the Coefficient of Friction.

The experiment of course shews that the friction acts along the surface opposite in sense to the force tending to cause motion.