The moment of P=P × OM sin OMB = Py sin A, the moment of QQ MB sin A = Q(h-x) sin A,

the moment of R=R× (BC—OM) sin A = R (k− y) sin A, the moment of S = S× AM sin A = Sx sin A.

Now if O be a point on the line of action of the resultant the algebraical sum of the moments of the forces round O vanishes, that is,

or

Py+Q(h-x)+R (k-y)+Sx=0,

(P-R)y-(Q-S) x + Qh + Rk=0.

This gives the relation which must hold between AM and MO, when O is a point on the line of action of the resultant.

1. ABCD is a square. A force of 3 lbs. acts from A to B, a force of 4 lbs. from B to C, and a force of 5 lbs. from C to D: find the single force which will preserve equilibrium.

2. A man carries a bundle at the end of a stick over his shoulder: as the portion of the stick between his shoulder and his hand is shortened, shew that the pressure on his shoulder is increased. Does this change alter his pressure on the ground?

3. If forces in one plane reduce to a couple, shew that if they were made to act on a particle, retaining their mutual inclinations, they would keep the particle at rest.

4. ABC is a triangle; H, I, and K are points in the sides such that

shew by taking moments round A, B, C that forces denoted by AH, BI, and CK are equivalent to a couple, except when H, I, and K are the middle points of the sides, and then the forces are in equilibrium.

5. A and B are fixed points; at any point C, in the arc of a circle described on AB as a chord, two forces act, namely, P along CA and Q along CB: shew that their resultant passes through a fixed point on the other arc which makes up the complete circle.

6. ABCD is a quadrilateral inscribed in a circle; if forces P, Q, R act in directions AB, AD, CA so that PQR as CD: BC: BD, shew that they are in equilibrium.

7. Two forces are denoted by MA and MB, and two others by NC and ND: shew that the four forces cannot be in equilibrium unless MN bisects both AB and CD.

8. Find a point within a quadrilateral such that if forces be represented by straight lines drawn from it to the angular points of the quadrilateral the forces will be in equilibrium.

9. Forces proportional to 1, 3, and 2 act at a point and are in equilibrium: find the angles between their lines of action.

10. If two equal forces P and P acting at an angle of 60° have the same resultant as two equal forces Q and Q acting at right angles, shew that P is to Q as √2 is to √3.

11. Cand B are fixed points; CA and CB represent two forces; if A move along any straight line shew that the extremity of the straight line which represents the resultant moves along a parallel straight line.

12. Forces denoted by the sides of a polygon, except one side, act in order: shew that they are equivalent to a single resultant which is parallel to the omitted side.

VII. Constrained Body.

99. A body is said to be constrained when the manner in which it can move is restricted. A simple example is that in which a body can only turn round a fixed axis, that is, can receive no other motion. In such cases forces may act on the body besides the restraints which restrict the motion, and we may require to know the conditions which must hold among these forces in order to ensure the equilibrium of the body.

100. When a body can only turn round a fixed axis and is acted on by a system of forces in a plane perpendicular to the axis, such that the algebraical sum of the moments of the forces round the point where the axis meets the plane vanishes, the body will be in equilibrium.

If the system of forces be not in equilibrium it is equivalent to a single resultant or a couple.

In the present case the system of forces cannot be equivalent to a couple; for then the algebraical sum of the moments would not vanish for any point in the plane.

Suppose that the system of forces is equivalent to a single resultant. Since the algebraical sum of the moments of the forces vanishes round the point where the axis meets the plane, the line of action of the resultant must pass through the point. Therefore the resultant has no tendency to turn the body round the axis; and the body is therefore in equilibrium.

101. The investigation of the preceding Article shews that the condition there stated is sufficient for equilibrium. The condition is also necessary for equilibrium; for if the condition does not hold, the system of forces is equivalent either to a couple or to a single resultant which does not pass through the axis, and in either case the body would be set in motion round the axis.

102. The most simple case of the preceding two Articles is that of the lever. A lever is a rigid body which is moveable in one plane about a point which is called the fulcrum, and is acted on by forces which tend to turn it round the fulcrum. In order that the lever may be equilibrium the moments of the two forces round the fulcrum must be equal and contrary, by Art. 101. Hence the condition of equilibrium stated in Art. 100 is often called the Principle of the Lever.

103. A body which is not constrained is called a free body. From considering the equilibrium of a constrained body we may render our conception of the equilibrium of a free body more distinct. Any condition which is necessary for the equilibrium of a constrained body will also be necessary for the equilibrium of a free body; although a condition which may be sufficient in the former case will not generally be sufficient in the latter case.

For example, in Art. 86 a certain principle is established with respect to the equilibrium of a free rigid body, and the investigation of Art. 100 shews us the interpretation of the principle. Suppose a body in equilibrium under the action of a system of forces in one plane. Imagine two points in the body, which lie in a straight line perpendicular to the plane, to become fixed. This cannot disturb the equilibrium, for we do not communicate any motion to the body by fixing two points in it; we merely restrict to some extent its possible motion. The body has still the power of turning round the straight line which joins the fixed points; and, by Art. 101, the body will not be in equilibrium unless the algebraical sum of the moments of the forces round the point where the straight line cuts the plane vanishes.

104. Suppose a body can only turn round a fixed axis, and that it is acted on by forces which are not all in one plane perpendicular to the axis; a strict demonstration of the condition of equilibrium is rather beyond our present range, but by assuming some principles which are nearly self-evident we shall be able to give a sufficient investigation.

First suppose the forces to consist of various systems in planes which are all perpendicular to the axis. It may be assumed as nearly self-evident that the tendency of the systems to set the body in motion will not be altered if all the other planes are made to coincide with one of them; and then the forces reduce to a system in one plane perpendicular to the axis, and Arts. 100 and 101 apply.

Next suppose the forces to be any whatever. Resolve each force into two components at right angles to each other; one component being parallel to the fixed axis. It may be assumed as nearly self-evident that the components parallel to the axis have no tendency to set the body in motion round the axis; and they may accordingly be left out of consideration.

The other components form various systems of forces in planes which are perpendicular to the axis; and, as in the first case, they may be supposed all to act in one plane, and Arts. 100 and 101 apply.

105. Suppose a body capable of moving only in such a manner that all points of the body describe parallel straight lines. For example, two fixed rigid parallel straight rods may pass through the body, and so the body be only capable of sliding along the rods. Suppose also that a system of forces acts on the body. Resolve each force into two components at right angles to each other, one component being parallel to the fixed rods. Then the necessary and sufficient condition of equilibrium is that the sum of the components parallel to the fixed rods, that is to the direction of possible motion, should vanish.

Hence we see the interpretation of the condition in Art. 90 relative to the equilibrium of a free rigid body.

106. When three forces maintain a body in equilibrium their lines of action must lie in the same plane.

Suppose a body in equilibrium under the action of three forces. Imagine two points in the body, one on the line of action of one force, and the other on the line of action of another force, to become fixed, the points being so taken