Elementary statics - Lock

It is hoped that this little book will be found to be a suitable text-book for students preparing for the Cambridge Previous Examination, for Woolwich, for the Oxford and Cambridge Certificate, for the London Matriculation, for the Local, and for other Examinations of a similar nature. At the same time I have endeavoured not to lose sight of the importance of the subject as an introditction to the study of Physics and of Practical Mechanics.

A knowledge of the ‘Trigonometry of one Angle’ is assumed in some parts of the book - it will be found, however, that considerable portions may be read without any acquaintance with Trigonometry. The truth of the Parallelogram of Forces is assumed, and the student who has not already read some elementary Dynamics is recommended to postpone the consideration of the proof until he reaches that subject.

I have therefore based the whole subject on Newton's Laws of Motion, a method which in my opinion greatly simplifies the subject. The accustomed proofs of the fundamental propositions based upon the principle of the Transmissibility of Force are given in a separate Chapter.

The use of the word Resolute, as the proper abbreviation for ‘Resolved Part’, will I hope be found useful in emphasizing the importance of the idea.

This is required in the Additional Subjects of the Cambridge Previous Examination. See Examination Papers at the end of this book. I have added a chapter on Graphic Statics and have reserved for that Chapter the consideration of the ‘Triangle of Forces’; as I venture to think that the method of solution based upon purely geometrical principles is best kept distinct from that based upon the Resolution of Forces.

The Examples have been made as simple as possible; the collection of 100 Miscellaneous Examples at the end of the book will be found somewhat more difficult.

For this SECOND Edition (which has been stereotyped), the whole work has been very carefully revised. Thanks to the assistance of many friends and teachers many defects have been removed.

The general character of the work is unchanged; but I have slightly enlarged its scope by the insertion of some illustrative problems worked out in pages 231 to 238, together with a carefully graduated set of interesting examples for exercise. I have also increased the miscellaneous examples at the end by the addition of problems selected and adapted from those set in Cambridge in the last two or three years.

Any corrections or suggestions will be gratefully received by the Author or the Publishers.

We propose in this Chapter to consider the nature of the stress set up between the surfaces of two separate rigid bodies which press the one on the other.

We shall confine our attention to the case of a fixed plane and a body placed upon it.

When a rigid body has one of its surfaces in contact with a fixed rigid plane, then, so long as the surface remains in contact with the plane, the only motion of which the body is capable is in some direction parallel to the plane.

For, whatever may be the forces acting upon the body, the plane, being rigid, can and does exert upon it whatever force is necessary to prevent motion in the direction perpendicular to the plane.

This force, applied by the plane to the body in the direction perpendicular to its surface, is equal and opposite to the perpendicular pressure of the body on the plane.

If there were a rigid substance whose surface could be made perfectly smooth, then the pressure applied by such a surface to any other surface pressing against it would be exactly in the direction at right angles to the smooth surface.

Although no such substance is known, yet it is often convenient to imagine such a substance for the purposes of theoretical statics. For the surfaces of some substances can be rendered very much more smooth than those of others, and hence we obtain results which are approximately true for surfaces which are comparatively smooth.

Moreover, for the purposes of explanation, we at first make our problems as simple as possible; advancing from the simple problems of theory to the more complex problems of practical mechanics.

Accordingly we use the words smooth and rough in the following technical sense.

DEF. A surface is said to be smooth which is understood to be incapable of exerting pressure on any other body in contact with it except in the direction perpendicular to itself.

DEF. A surface is said to be rough which is capable of exerting pressure on a body in contact with it in other directions besides that perpendicular to itself.

It will be seen that a smooth surface may be said to be one which offers no resistance to the motion along it of a body which is pressing against it.

A rough surface does offer some resistance to the motion along it of a body pressing against it.

For all actual surfaces, even those that appear to be perfectly plane, would if sufficiently magnified be seen to consist of minute projections and depressions as in the Figure. These always interlock more or less with another surface in contact with it, and oppose resistance to a force such as the horizontal component of /"which tends to slide one surface over the other.

DEF. A perfectly rough surface is one which is capable of offering whatever resistance may be necessary to prevent the motion along it of a body pressing against it.

Thus a perfectly rough surface is a theoretical surface along which bodies pressing on it cannot move.

It is as we have said impossible to get a surface smooth as defined above. It must however be understood that when we speak of a 'rough' plane surface we mean a surface which has been made as nearly smooth as the nature of the substance will permit.

For instance, if the material is of metal it is understood that the surface has been carefully polished; if of wood, that it has been planed; if of stone, that it has been rubbed down, and brought to as smooth a surface as the particular kind of stone will allow. Its surface will then depend only on the material; it will be smooth to the eye and touch, but not statically smooth.

Consider now the action exerted by a fixed 'rough' plane upon a body which presses against it.

Since the plane is rough, the force applied to the body is not necessarily perpendicular to the plane.

We resolve this force into two rectangular components, one at right angles to the plane, the other along the plane.

The component force at right angles to the plane is called the pressure between the plane and the body.

The component force along the plane is called the friction between the plane and the body.

The friction is the part of the action of the plane on the body which prevents the movement of the body along the plane.

When a body is at rest on a fixed plane the action of the plane upon it is equal and opposite to the resultant of all the external forces acting on the body. The friction is therefore equal and opposite to the part of the external forces which tends to cause motion of the body on the plane. Hence we have the following definition.

DEF. When a body rests on a rough plane and forces act on the body tending to cause it to slide along the plane, a force is called into play which acts on the body in the direction contrary to that in which it tends to move, and tends to prevent motion.

This force is called Friction.


The General Laws of Friction are

I. When the surfaces of two bodies are in contact and at rest relatively to each other friction always acts upon each body in the direction exactly opposite to that in which the body tends to move along the surface on which it presses.

II. When there is no relative motion, the amount of friction which is called into action is just as much as is necessary to prevent relative motion - and no more.


The mass is under the action of the following forces:

(i) its own weight downwards,

(ii) the pressure of the plane on it upwards,

(iii) the force P horizontally,

(iv) the force of friction, acting between the surface of the mass and of the plane; this force is horizontal and must be equal to the only other force acting, namely P lbs., and must act in exactly the opposite direction.

If the plane in the present case were perfectly smooth, in other words, if we suppose friction not to exist, it is clear that the force P would cause the mass to move in its own direction.