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Equilibrium of Forces

1. Torque. — The tendency of a force to produce rotation depends on the magnitude of the force and on the perpendicular distance between the line of action of the force and the axis about which the rotation takes place. It is proportional to the magnitude of the force and also to the distance between the line of action of the force and the axis of rotation. It is convenient to define torque, or moment of force, as the product of the force and the perpendicular distance between the line of action of the force and the axis of rotation.

Torque = force Х distance from axis

2. Conditions of Equilibrium. — In order to make a body in equilib­rium under the action of any number of forces, two conditions must be satisfied:

1. The sum of the forces acting on the body in any direction must
be equal to zero. When this condition is satisfied, the body will have
no tendency to change its motion of translation, since there is no net force acting on it.

2. In order that the body may have no tendency to change its mo­tion of rotation, the sum of the moments of force tending to produce clockwise rotation about any axis must be equal to the sum of the moments of force tending to produce counterclockwise rotation about that same axis. When this second condition is fulfilled, there is no net torque acting on the body, and its motion of rotation will not change with time, i. e., if the body is already at rest, it will not start into rotation; and, if it is already in rotation, its rate of rotation will not change.

3. Center of Gravity. — Every particle of a body possesses weight, so that the pull of the earth on the body is made up of a large number of forces directed toward the center of the earth.

Suppose that there are two particles of mass, m and M, at A and B, respectively, and that these particles are connected by a light rod. These particles are attracted to the earth with forces that are nearly parallel to each other. If a point С in the rod is so chosen that

m Х AC = M Х BC

then the moments of force tending to turn the rod clockwise are just equal to the moments tending to turn it counterclockwise. If the rod is turned into some other position, the forces will no longer be perpendicular to the rod, but the moments of force about С will still balance each other. Hence, it is possible to regard the two masses as concentrated at C, since the action of gravity on these two masses concentrated at С is the same as its action when the masses are at the ends of the rod. A point at which it is possible to assume the masses concentrated without changing the action of gravity on them is called the center of gravity of the masses.

Whatever the shape and size of the body, it is always possible to find one point at which a force equal and opposite to the weight of the body can be applied so that the body will remain at rest.

About this point the body has no tendency to rotate under the action of gravity, and at this point we may consider all the mass of the body to be concentrated. If the body is balanced on a knife edge, this point will lie directly above the knife edge. The center of gravity need not necessarily lie in the substance of the body. Thus the center of gravity of a uniform ring lies outside the material of the ring - at its center.

4. Types of Equilibrium. — The equilibrium of a body may be stable, unstable, or neutral. When a body returns to its original position after being slightly disturbed, the equilibrium is said to be stable. A cone standing on its base is an illustration of this type of equilib­rium. When a cone on its base is raised slightly from the table on which it rests, it returns to its original position on being released. It is in stable equilibrium. If, however, the cone rests on its vertex and is then slightly displaced, it tends to fall into a new position rather than return to its original position. In this case the cone is in unstable equilibrium. Any body which tends to get as far as possible from its original position when disturbed is in unstable equilibrium. A sphere resting on a horizontal table when slightly displaced tends neither to return to its former position nor to go still farther away from it, but it remains in any position in which it finds itself. Such a body is in neutral equilibrium.

5. Stability of a Body. — The position of the center of gravity is of much importance in determining the stability of a body. The lower the center of gravity, the greater the stability of the body and the more difficult it is to overturn. A body becomes unstable as soon as the vertical line through the center of gravity falls outside its base. The body which must be displaced the greater amount in order to make the vertical through its center of gravity fall outside the base is the more stable [2, С. 48 - 49].

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