Rigid Body

A rigid body is a simplified version of a solid body that does not deform. In other words, regardless of external forces acting on a rigid body, the distance between any two points remains constant over time.

An example of a rigid body is a metal rod.

When a rigid body is in pure rotational motion, all of its particles rotate at the same angle and for the same amount of time. As a result, all particles have the same angular velocity and acceleration.

Rigid body

A rigid body (also referred to as a rigid object) is a solid entity having zero or minimal deformation, such that it may be ignored in physics. Regardless of external forces or moments acting on a rigid body, the distance between any two given locations remains constant throughout time. A rigid body is generally perceived as a continuous mass distribution.

A rigid body does not exist under special relativity, and things may only be assumed to be rigid if they are not travelling nearly at the speed of light. A rigid body is commonly thought of, in quantum mechanics, as a collection of point masses. Molecules, for example, are frequently seen as rigid bodies since they comprise point masses such as electrons and nuclei (see the classification of molecules as rigid rotors).

Equilibrium of a rigid body

A rigid body is in mechanical equilibrium when it neither changes its linear momentum nor its angular momentum at a given point of 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 and the vector of the torques are both 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 in time, and the body is said to be in translational equilibrium. When the total torque of the rigid body is zero, then its angular momentum does not change with time, and it is said to be in rotational equilibrium.

Equilibrium can be classified as static and 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.

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.

Centre of mass of a rigid body

Even though the centre of mass and centre of gravity often coincide, they are different. When the entire system is subjected to uniform gravitational fields, both the centre of gravity and centre mass will be the same. The shape of the object affects the centre of mass of a rigid body formula too.

Centre of mass formula for pointer objects:

zcom=i=1kmizimi

mi is mass of ith object, zi is distance from the z-axis of ith object

Moment of inertia

The moment of inertia is the resistance to the change in angular acceleration of the body. It is the sum of the product of the mass of every particle of the body and the square of the distance at which the object is placed from the rotating axis.

In simpler words, it can be said it is the quantity that decides the amount of torque required for acceleration.

Moment of inertia formula

The moment of inertia is the value due to the resisting angular acceleration and is the summation of the product of the mass of each particle with the distance square.

So, in simple words,

Moment of inertia, I = m × r2

Moment of inertia of rigid bodies

The moment of inertia of rigid bodies can be calculated by integrating. If the system of rigid bodies is divided into an infinite number of particles, then its mass, ‘dm’ and distance of mass from the axis of rotation is ‘r’. Now, the moment of inertia becomes,

I = ∫ r2 dm

Torque

In rotational motion, torque, i.e., the moment of force, is the ability to rotate an object rotating at a fixed axis. Mathematically, it can be calculated as per the below formula:

= rF = rFn

Here,  is the torque,

F is the force, r is the perpendicular distance, and is the angle between vectors F and r.

Radius of gyration

An object’s radius of gyration is defined as the root mean square of the distance of the particles from its rotational axis. It is usually written as K in mathematical formulas and relations. It can be given by the below equation:

K =r12+r22+…………+rn2n

If we multiply an object’s mass with the radius of the gyration square it is equal to the rotational inertia of the

object.

Therefore, I = MK 2

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.