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Rotational equilibrium is a state where a body, despite being subject to potential forces, does not rotate about any axis. To achieve this state, the conditions are straightforward yet crucial. The net external torque applied to the body must be zero. This is represented by the equilibrium equation for rotation, which can be mathematically stated as \( \sum \tau = 0 \), where \( \tau \) denotes the torque. It is important for the sum of clockwise torques to be perfectly balanced by the sum of counterclockwise torques. This balance prevents any angular acceleration, thus maintaining rotational equilibrium. An intuitive way to visualize this is by imagining a wheel that stays still despite various forces being applied around its rim because these forces cancel each other out.

While rotational equilibrium deals with the absence of rotation, translational equilibrium ensures that a body does not accelerate linearly; in other words, it does not start moving. The key condition here is that the vector sum of all external forces acting on the body must equal zero, leading to the equation \( \sum \mathbf{F} = 0 \). When dissected, this means that forces acting in one direction are counteracted by equal forces in the opposite direction. This concept is easily applied in everyday scenarios, such as a book resting on a table where the force of gravity pulling it down is exactly matched by the table's upward support force.

Torque is a measure of the force causing an object to rotate about an axis, and it is strongly dependent on both the magnitude of the force and its application radius from the axis of rotation. External torque specifically refers to the torque caused by forces that act outside the system being considered. For example, in a physics problem involving a seesaw, the weights of the children on either end create external torques about the pivot. It is calculated using the equation \( \tau = r \times F \sin(\theta) \), where \( r \) is the distance from the pivot point, \( F \) is the magnitude of the force, and \( \theta \) is the angle between the force vector and the displacement vector from the point of rotation. Balancing these external torques is essential for maintaining equilibrium.

In the realm of physics, the conditions for equilibrium require that two main criteria be satisfied consecutively; these underpin the static balance of any object. Analyzed separately, for translational equilibrium, all forces acting upon a body must counterbalance each other, inhibiting any form of linear displacement. On the other hand, for rotational equilibrium, the moments of the forces, which are the torques, must be in a similar equilibrium state ensuring no angular motion unfolds. It is not enough that a body is free from translational movement to consider it in complete equilibrium; it must also be devoid of any rotation. The synchronization of both translational and rotational equilibrium conditions is compulsory for a body to be truly deemed 'at rest' in the physical sense.