Equilibrium Of Rigid Bodies

To understand equilibrium in a rigid body, it is first essential to know what a rigid body is and an equilibrium point. 

A rigid body is a substance that has a definite shape that does not change with external force. A rigid body is a system of particles that are each distanced equally, and the distance cannot be changed.

A body or particle has an equilibrium state if the motion or internal energy of the particle doesn’t change over the time. For example, body equilibrium is when the total external force or torque is zero.

Equilibrium of rigid bodies

A rigid body is in mechanical equilibrium when it is neither changing its linear momentum nor its angular momentum at a given point in time. In the absence of linear and angular acceleration, the body is in a state of mechanical equilibrium. 

In such a case, the vector sum of forces on the body is computed as zero, and the vector of the torques is also computed as zero.

When the total force on a rigid body is zero, then the linear momentum of the rigid body remains unchanged at a given point of time, and the body is said to be in translational equilibrium. When the total torque of the rigid body is zero, then the angular momentum of the rigid body does not change with time, and it is said to be in rotational equilibrium.

Equilibrium of rigid bodies can be classified as static and dynamic equilibrium.

Dynamic equilibrium:

When the body is in motion and continues being in motion at a uniform velocity, it is said to be in dynamic equilibrium.

Static equilibrium:

When the body is at rest and continues to be at rest without any change in the motion or momentum of the particles, it is said to be in static equilibrium or Equilibrium of rigid bodies.

General conditions for equilibrium of a rigid body:

Sometimes a rigid body can be in partial equilibrium. When a rigid body is in partial equilibrium, it shows either translational or rotational equilibrium.

For example, consider a rigid stick placed under the influence of external force. 

When the stick is in rotational equilibrium then a force of equal magnitude is applied in the opposite direction to two ends of the stick, so there will be translational motion.

When a pair of forces are equal in magnitude, but opposite in direction and different in their line of action, the pair is called couple or torque. A couple can produce rotational motion but not translational motion.

Centre of Gravity:

The Centre of gravity is where a rigid body is balanced and results from a mechanical equilibrium between two rigid bodies. For example, if a scale is balanced on the tip of a finger, then the tip of the finger acts as the point of balance for the scale and is referred to as the centre of gravity for the scale. 

The scale is balanced at the tip of the finger when the tip has an equal and opposite force to the force of gravity. At the point of balance, the total torque is zero due to gravity’s force, which keeps the scale in equilibrium on the tip of the finger.

Conclusion:

Equilibrium of rigid bodies does not imply that the body is doing nothing. It is a state in which a process occurs in the body yet a balance is achieved, resulting in no change in the body’s state. It’s a state in which neither the body’s status nor its internal energy changes.

If the three conditions below are met, a solid body subjected to three forces with non-parallel lines of action is in equilibrium:

• The action lines are parallel (in the same plane)
• The action lines are converging (they cross at the same point)
• The zero vector is equivalent to the vector sum of these forces.
• The forces are confined in the plane Pxy in the animation. P is the intersection point, and the vector sum of the forces is always zero. The mass has reached a state of equilibrium.

As long as the sum of the forces acting on the body stays zero, the body is in equilibrium. “Unless acted upon by a force that would cause them to change their state of motion, all bodies will continue in their state of rest or uniform motion along a straight line that they are now in,” Newton’s Law of Inertia states.

Statics is the branch of mechanics that investigates solids at rest.