Equilibrium

We have already used this term in reference to balanced force system. We used the concept of equilibrium with an implicit understanding that the body has no rotational tendency. In this module, we shall expand the meaning by explicitly considering both translational and rotational aspects of equilibrium.

A body is said to be in equilibrium when net external force and net external torque about any point, acting on the body, are individually equal to zero. Mathematically,

$$\Sigma F=0$$

$$\Sigma \tau =0$$

These two vector equations together are the requirement of body to be in equilibrium. We must clearly understand that equilibrium conditions presented here only ensure absence of acceleration (translational or rotational) – not rest. Absence of acceleration means that velocities are constant – not essentially zero.

Now, we study translational motion of rigid body with respect to its center of mass, the linear and angular velocities under equilibrium are constants :

$$v_C=\text{constant}$$

$$\omega =\text{constant}$$

We need to analyze equilibrium of a body simultaneously for both translational and rotational equilibrium in terms of conditions as laid down here.

Equilibrium types

We are surrounded by great engineering architectures and mechanical devices, which are at rest in the frame of reference of Earth. A large part of engineering creations are static objects. On the other hand, we also seek equilibrium of moving objects like that of floating ship, airplane cruising at high speed and such other moving mechanical devices. In both cases – static or dynamic, external forces and torques are zero.

An equilibrium in motion is said be “dynamic equilibrium”. Similarly, an equilibrium at rest is said be “static equilibrium”. From this, it is clear that static equilibrium requires additional conditions to be fulfilled.

$$\Rightarrow v_C=0$$

$$\Rightarrow \omega =0$$

Equilibrium equations

In general, a body is subjected to sufficiently good numbers of forces. Consider for example, a book placed on a table. This simple arrangement actually is subjected to four normal forces operating at the four corners of the table top in the vertically upward direction and two weights, that of the book and the table, acting vertically downward.

If we want to solve for the four unknown normal forces acting on the corners of the table top, we would need to have a minimum of four equations. Clearly, two vector relations available for equilibrium are insufficient to deal with the situation.

We actually need to write two vector equations in component form along each of thee mutually perpendicular directions of a rectangular coordinate system. This gives us a set of six equations, enabling us to solve for the unknowns. We shall, however, see that this improvisation, though, helps us a great deal in analyzing equilibrium, but is not good enough for this particular case of the book and table arrangement. We shall explain this aspect in a separate section at the end of this module. Nevertheless, the component force and torque equations are :