Dynamics of rigid bodies with fixed axis of rotation

Introduction

When a rigid system of particles or a rigid body is in motion, each particle follows a particular path. The path of the particles moving depends on the kind of motion the body experiences. However, the movement of particles is different when the body is in translational motion than in rotational motion; in rotational motion, factors like dynamics of rigid bodies with fixed axis of rotation influence the particle behaviour. 

When rotation is along a fixed axis, all the particles in the body rotate along one fixed axis. However, while in rotational motion, a body experiences multiple factors that come under the dynamics of rigid bodies with a fixed axis of rotation, such as angular velocity, the moment of inertia, and Torque. 

Fixed Axis Rotation

When a body or a system experiences a rotational moment along with a fixed point in space, it is called fixed axis rotation; as the body and its particles move in rotational movement in a circular path, every particle in the body moves in a different path of different radius. However, all the particles share the same axis.

There can be plenty of examples for dynamics of rigid bodies with fixed axis of rotation; some of the most visible examples are ceiling fans, wind turbines, a disk in a CD player, and a carousel. 

Components in Fixed Axis Motion

In the dynamics of rigid bodies with fixed axis rotation, the perpendicular component of torque is not considered; only those components that work along the direction of motion are considered. It is because only those components of Torque that are applied along the direction cause the body to rotate here; the perpendicular component doesn’t exist. The forces parallel to the axis apply the torque perpendicular to the axis and are ignored.

Rigid Body Dynamics

The dynamics of rigid bodies with a fixed axis of rotation can be understood by the relationship between work done by the torque on the rotating particles in a moving circular path. Consider the particles of the body in motion on a circular path with a centre Q on an axis. Consider the P1P’1 arcs created by a small displacement ds1. We have learned that in the dynamics of the bodies with a fixed axis of rotation, forces perpendicular are considered.

Consider, at a specific time, force F1 working on a particle P1 and also lies on a plane perpendicular to the axis of rotation.

This plane is called the x’-y’ plane, and particle P1 follows a path with radius r1.

Here, QP1= r1

And particle P1 moves to P’ in time 🛆t .ds1, (displacement of the particle)

ds1 = r1 d

dӨ is the angular displacement of the particle total work done 

d W1= F1 ds1 cos ɸ1=F1(r1d) sin 1

ɸ1= angle between the tangent at P1 and force F1.

the angle between the radius vector and F1= 1

Torque

The Torque is the rotational force component; force causes linear acceleration; Torque causes angular acceleration. The pair can also be described as the “transformation capacity” of a force on an axis.

 τ= QP F1

τ= r1 F1

Work done by different forces

Let’s consider a body where more than one force acts on it.

In a body with more than one force acting, the work done is the total sum of work done by each; this gives the total work done by the body. 

The magnitude of torques τ1, τ2,τ3…… etc. 

So, 

 dW = (τ1+ τ2+τ3 +……)dθ.

dθ will remain the same. 

We have already considered that all the torques parallel to the fixed axis. 

And the total amount of the total external force is the sum of individual torques by various particles. 

we get,

 dW = τ dθ.

The obtained equation gives the total work done by Torque, acting on the rigid body by turning a fixed axis.

Conclusion

Dynamics of rigid bodies with a fixed axis of rotation include multiple factors, including angular velocity, angular acceleration, a moment of inertia, and Torque. The rigid bodies are made of a system of particles; when such bodies are in motion, all of their particles also experience movement. However, these particles move with the same uniform velocity. In a rotating movement, the rigid body particles move in different paths of different radii. 

However, all the body particles share the same centre or axis of rotation. In a practical sense, we may come across various examples of dynamics of rigid bodies with a fixed axis of rotation. Some notable examples can be a ceiling fan, a wind turbine, and a merry-go-round in a theme park.