Rotational Dynamics

We know dynamics is a branch of physics that deals with the physics associated with the objects or particles in motion. The dynamics have been subdivided into various categories depending on the subject of interest. For example, the motion of the fluids is studied under fluid dynamics, the motion of electrical charges are studied under electrodynamics, similarly one of these categories is rotational dynamics. Rotational dynamics is a branch of physics that deals with the forces and motion about an axis of rotation. 

In this article, we will study the system of particles and rotational dynamics with a brief introduction to the system of particles and rotational motion notes. Let’s begin!!

Archimedes of Syracuse

Since Archimedes' time, the concept of rotational dynamics, and specifically the moment of force or torque associated with rotational dynamics, has been researched. You may recall Archimedes' famous phrase, "Give me a lever, and I shall move the globe," from your school days. A lever is used to apply torque. The lever is employed to put rotational force on a threaded fastener in this situation. 

Archimedes of Syracuse (c. 287 – c. 212 BC) was a Greek mathematician, physicist, engineer, astronomer, and inventor who lived in the ancient Sicilian city of Syracuse.  Despite the fact that nothing is known about his life, he is recognised as one of classical antiquity's foremost scientists.

System of Particles and Rotational Motion Notes

In rotational dynamics physics, Newton’s Second Law of motion is used to develop a formalism to describe how objects rotate. In other words, rotational dynamics introduce the concept of torque which represents a comparable role to that of force in non-rotational dynamics. In rotational dynamics, we will study the concept of torque, a moment of inertia to describe how objects oppose the rotational motion.

When a particle or an object is subjected to a motion in a straight line then the motion is known as translational motion, at the same time when the same particle moves along an axis in a circular path then it is known as Rotational Motion and the branch of physics that deals with the rotational motion are known as rotational dynamics. The main concepts of rotational dynamics are the system of particles and rotational motion. For example, consider a ring rolling without slipping in a straight line. The forth displacement of the ring is equal to the linear displacement of a point fixed on the rim.

We often refer to an object as an extended body in rotational dynamics, it implies that we are referring to it as a system of particles. Rigid bodies are those bodies with definite shape and size. In rigid bodies, the distance between the constituting pairs of particles does not change.

Torque: Rotational Force

During the linear motion, an object which is subjected to linear motion will move till it feels any external force acting on it. The external force acting will change the direction of motion of objects. However, when a body is moving about a fixed axis in any circular path or curved path then, it will also feel the force and the force acting on the bodies under rotational motion is known as the Torque. It is always better to understand with an example, we notice that it is always easy to open a door by pushing on the edge farthest from the joints than by pushing in the middle. Therefore, torque is a force that plays an important role in rotational dynamics.

To develop the precise relationship among force, mass, radius, and angular acceleration, consider what happens if we exert a force F on a point mass m that is at a distance r from a pivot point, as shown in Figure. 

Because the force is perpendicular to r,  an acceleration a is obtained in the direction of F According to newton’s laws of motion force is equal to the product of mass and acceleration i.e.,F = ma. We know that the relation between the linear acceleration(a) and rotational acceleration(α) is given by a = rα, and we substitute this value of a into F=ma, we get:

$\Rightarrow F = mr\alpha$……(1)

Where,

a - linear acceleration

α-rotational acceleration

Now, we know that the physical quantity torque is the changing effectiveness of a force. Therefore, since the force F is perpendicular to r, torque (τ) can be simply written as  

τ = Fr.

Therefore, multiplying equation (1) on both sides by r, we get:

$\Rightarrow F=mr\alpha$

$\Rightarrow rF=mr^2\alpha$

$\Rightarrow \tau=mr^2\alpha$….(2)

Equation (2) is known as Newton’s second law of motion in rotational dynamics, where the torque is analogues to force and angular acceleration is analogous to linear or translational acceleration and the term mr2 is analogous to mass. The product of mass and square of the distance from the axis of the rotation is known as the moment of inertia.

The best examples of rotational motion are rigid bodies. The rigid bodies experience both translational and rotational motion. Thus, for rigid bodies, both the linear velocity (corresponding to translational motion) and the angular velocity (corresponding to rotational motion) need to be analyzed. These problems can be simplified by defining the translational and rotational motion of any rigid body separately.

The System Of Particles And Rotational Dynamics is completely analogous to linear or translational dynamics. Most of the equations that are derived for the mechanics of rotating objects are quite similar to the motion equations derived for linear motion. In System Of Particles And Rotational Motion, particularly rigid bodies are considered. A rigid body is an object with a mass that holds a rigid shape.

For a system of particles, the centre of mass is described as that point at which the entire mass of the system is imagined to be concentrated, for consideration of its translational motion. If all the external forces acting on the system of particles were to be applied at the centre of mass, the state of rest or motion of the system of particles and the rotational motion shall remain unaffected.

The centre of mass of a body is a balancing point for any system of particles. The centre of mass of a two-particle system is always found to be on the line joining the two particles and it is often located in between the particles.

Moment of Inertia and Rotational Motion

Now, let us understand the important concept of rotational motion and rotational dynamics, i.e., Moment of inertia. The moment of inertia of a rigid body is often defined with respect to a fixed rotation axis, it is a physical quantity that determines the amount of torque (Force acting on bodies under rotational motion) required for obtaining required angular acceleration or a physical property of a rigid body due to which it resists angular acceleration. 

The formula for the moment of inertia of a rigid body is given by the sum of the product of the mass of each particle with the square of rigid body distance from the axis of the rotation. Mathematically, the moment of inertia formula is given by:

$I=\sum_{i=1}^nm_ir_i^2$

Where,

• m -The mass of the rigid body

• r - The distance from the axis of the rotation

• i - The number of particles

Similarly, we study the rolling motion, angular acceleration, angular velocity, theorems related to the moment of inertia, etc… in the system of particles and rotational motion notes or rotational dynamics notes.

Solved Example

1. In a car garage loosens a bolt by applying a force of 500 N to a wrench. Further, the man applies the force such that the applied force will be perpendicular to the arm of the wrench. The total distance of the bolt to the man's hand is about  0.50 m. Calculate the magnitude of the torque produced?

Sol: 

Given,

The amount of force applied to the bolt = F = 500 N

The angle between the applied force and the wrench θ = 900 

The distance between the wrench and the arm of man = r = 0.50 m

Now, we are asked to determine the magnitude of the torque produced while loosening the wrench. We know that the Torque can be calculated using the following equation:

$\tau=F\times r \times \sin \theta$

$\tau=500\times 0.5 \times \sin 90^o=250 Nm$

Therefore, the magnitude of the torque produced is 250 Nm.

Interesting Facts:

Torque is an important factor in fidget spinners!! Yes, the technology of fidget spinners evolved earlier than they ever began spinning. Look at it. It simply sits there. Even while it seems to be doing not anything at all, the spinners are exemplifying essential physics concepts. If you need the item to rotate you need to exert a torque at the spinner. 

Torque relies upon the importance of the carried out pressure, how a way from the rotation axis the pressure is carried out, and what sort of the pressure is perpendicular to the arm of the spinner. You'll want to push it with loads of oomph as far from the middle as possible, and at the least barely withinside the path you need it to spin.

Conclusion

In this article we understood the dynamics of rotational motion and the importance of the Torque in rotational motion.