Equilibrium of Rigid Bodies: Mechanical, Partial

If the net external force or torque on a particular point in a system of particles is zero, it is said to be in equilibrium. These points of equilibrium may lie anywhere around the center of mass of the body. For any object in translational motion, external force results in the change in the linear momentum, whereas for the object in rotational motion, external torque applied produces the change in the angular momentum.

When the object into consideration (system of particles) is in the state of mechanical equilibrium, the linear and angular momentum remains conserved with time, i.e. unchanged. We can also conclude from this that the body without any external force does not possess linear or angular acceleration. Hence, we can conclude that:

• The absence of net force on a rigid body results in conservation of linear momentum and hence, puts it in translational equilibrium
• The absence of net torque on a rigid body results in conservation of angular momentum and hence, puts it in rotational equilibrium

Mechanical Equilibrium: An overview

In general, all the forces on a rigid body act in the same plane, i.e. they are coplanar. The body attains translational equilibrium when the sum of any two of the components along any of the perpendicular axis is zero. The condition of translational equilibrium is dependent on the particle nature taken into consideration, hence, the vector sum of all the external forces acting on the particle must be zero. On the other hand, the body attains rotational equilibrium, when the sum of all three components is zero.

Partial Equilibrium: An overview

For any rigid body, the state of equilibrium is not always complete equilibrium, sometimes it is partial equilibrium as well. When a rigid body shows a single type of equilibrium, it is said to possess partial equilibrium.

For example, if on a wooden plank, pivoted from the middle, the two forces of equal magnitude acting on it in the opposite direction tend to rotate it. The net moment of the entire system sets zero and the plank is set in rotational equilibrium. If the forces at the ends of planks are opposite and act in the direction perpendicular to the rod, the translational equilibrium comes into play. These forces are actually not opposite, rather, these play in the same sense, resulting in setting up anti clockwise rotation in the rod, keeping net external force to zero. Such equal and opposite forces which act in different lines of action, sets the body (wooden plank in this case) into rotation without actually setting it up in translation are termed as couples. Couple as an external force sets the body in motion, hence it is also termed as torque.

In the example mentioned above, the wooden plank was resting on the fulcrum exactly at the center of the plank. To understand mechanical equilibrium best, we need to understand the lever-fulcrum system with varying positions of fulcrum.

Consider a wooden plank-fulcrum system where fulcrum is not situated in the middle. Here, the reactive force R, acting in the direction perpendicular to the lever, at the point of fulcrum. This force acts in the direction opposite to the direction of the couple forces F₁ and F2, making R – F₁ – F₂ = 0. When such a situation is visible, the rigid bodies are said to attain translational equilibrium. On the other hand, rigid bodies attain rotational equilibrium, when D₁F₁ – D₂ F₂ = 0, where D₁ and D2 are respective distances of the load F₁ and effort F₂ from the fulcrum point, i.e. the sum of moments about the fulcrum point should be zero. D₁ here is termed as load arm and D₂ is termed as effort arm. When rotational equilibrium is attained D₁F₁ becomes equal to D₂ F₂ .

• F₁F₂ = D2D₁

Here F₁F₂ is known as mechanical advantage which depends on D₁ and D₂ .

When D₂ is greater than D₁, the mechanical advantage is greater than 1.

This implies, even with a small amount of effort, a large quantity of load can be lifted.