Equilibrium of a rigid body

Introduction:

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.

Rigid body in motion:

Understanding the equilibrium of a rigid body will require one to understand the motions that a rigid body can undergo. 

The two motions that a rigid body can undergo are:

Translational Motion: The motion where all the particles of the rigid body move in a linear direction; it is known as the translational motion of the rigid body. Whereas a pure translational motion is a motion where the rigid body not only moves along a linear plane, but all particles of the body have the same velocity at a given point of time.

Imagine a box being pulled on horizontal ground. The box is moving along the linear plane and has no sideward movement, and all particles of the box are moving in the same direction and at the same velocity at a given point time. In such a case, the rigid body, which is the box, in this case, is in pure translational motion.

If we imagine a ball rolling over the same ground, due to the ball’s spherical shape, all the particles will not have the same velocity at a given point of time, but they will still move through the linear plane. This motion of the ball is not a pure translational motion.

What happens when a rigid body is restricted to a translational motion, i.e., moving linearly? The answer is the second kind of motion – the rotational motion.

Rotational Motion: When a rigid body is fixed along a line or point, the only possible move that the rigid body can achieve is around the fixed point, and such a motion is called the rotational motion. The fixed line along which a rigid body rotates is the axis of rotation. The most common example of this motion is the earth’s rotation around its axis or a fan. In rotation, a rigid body and all the particles in the body moves in a circular motion centred at the axis.

When a ball rolls on the ground (as discussed in an earlier example), the ball rotates along with a fixed point and moves forward on a linear plane. During this, the ball particles are moving in a circular motion centred at the axis and moving forward in a linear direction. Thus, the ball is in both translational and rotational motion in this scenario.

When a rigid body is in motion and is not fixed to a point, then it is either in translational motion or in both translational and rotational motion. However, it is rotational if a body is in motion and fixed to a point or line.

Equilibrium of a rigid body:

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 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

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.

From the above diagram: When force is applied on two ends of the stick in the perpendicularly opposite direction of the sticks, then it causes a rotational motion, and there is no translational motion to the stick. Thus, the stick is in translational equilibrium.

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:

Thus, a rigid body is a system of particles that are distanced equally, and the distance cannot be changed. Of course, in the real world, we do not have any perfectly rigid bodies as all bodies change by an external force, but in some cases, the change is so negligible that it is considered a rigid body. Some examples are earth, metal balls, etc.

Reaching equilibrium in rigid bodies requires more than one force acting in the opposite direction to reach a state where the body does not experience linear and angular momentum. When the body is in equilibrium, the vector sum of forces or torques is zero, and the linear and angular momentum does not change at a given point in time.