## Acta Scientific Computer Sciences

#### Case Study Volume 3 Issue 6

Analysis of Inertial Torque Acting on a Rotating Body

Ryspek Usubamatov*

Kyrgyz State Technical University After I. Razzakov, Bishkek, Kyrgyzstan

*Corresponding Author: Ryspek Usubamatov, Kyrgyz State Technical University After I. Razzakov, Bishkek, Kyrgyzstan.

Received: April 15, 2021; Published: May 06, 2021

#### Abstract

The mathematical model for the rotation of a body about a fixed point is a classical topic of physics mechanics. Physics considers planar rigid-body motion that is presented by rotation about a fixed point and curvilinear motion. In both cases, the particles of the rigid body move along the circular path and have their velocities and accelerations. Analysis of the motions demonstrates the particles subjected by the radial and angular accelerations. The physics mechanics does not describe the angular acceleration of particles and is confined only by the radial one. The angular acceleration of particles has the same right as other fundamental principles of physics. Engineering practice requests exact computing of accelerations of the mechanical components for the quality of machine work. This short communication considers the physical interpretation of the accelerations of a rotating body, its turn around own center mass, and inertial torque acting on it.

Keywords: Radial and Angular Acceleration; Inertial Force and Torque; Rotating Body

## Introduction

All textbooks of engineering mechanics contain the chapter that considers the rotation of the body with the constant angular velocity about a fixed axis and about the fixed point when the body‘s axis is offset on some distance r [1-7]. The mathematical model for the rotation of the body about a fixed axis is a follows: Jε = T where Jb (kg·m2) is the mass moment of inertia of the body, ε (rad/s2) is the angular acceleration, and T (N·m2) is the external torque acing on the rotating body. The left side of this expression Jε presents the inertial torque.

Analysis of the rotating body about the fixed point yields the radial inertial acceleration and hence the radial force that depends on the values of the angular velocity and radius of rotation.

The radial acceleration of rotating body about the fixed point is presented by the following equation a = rω2 where a is the radial acceleration, r is the radius of rotation of the body relatively of a fixed point of rotation and ω is the constant angular velocity of its rotation about a fixed point and the action of the external torque [8-15]. The left and right of the expression is components of the centrifugal and centripetal forces respectively according the textbooks. The textbooks present the centrifugal force that is Fct = ma (kg·m/s2), where m is the mass and a is the radial acceleration of the rotating body about the fixed point. The centripetal force is presented by the following: Fcp = mrω2 (kg·m/s2), where all components are as specified above. Two components of the expression have different physical interpretations, i.e., there is dual presentations of the one physical term. Such dualism contradicts mathematical logic and mathematics that is exact science and does not allow ambiguous interpretations.

Methodology

Analysis of the radial acceleration (a = rω2) yields the following: the angular velocity ω presents the scalar product of the angular velocity that express the angular acceleration, i.e., ω2 = ε. This angular acceleration does not relate to the rotation of the body about the fixed point as far as it rotates with the constant angular velocity ω about the fixed point. Logically, the angular acceleration ε relates only to the rotation of the body about its center mass that locates on the distance r from its fixed point of rotation. The angular acceleration ε of the body means it rotates about its center mass and subjected by the action of the external torque that expresses by the following: T = (J + mr2)ε, where J is defined by the parallel axis theorem and presented by the following: J = Jb + mr2 where Jb is the mass moment of inertia of the body about its center mass, other parameters are as specified above.

The rotating body about the fixed point is subjected to the action of the centrifugal force of the radial direction and the inertial torque that turns the body about its center mass. This inertial torque was missed from consideration by the physics that presents a fundamental principle of classical mechanics. All rotating bodies always turn around their center mass with the angular velocity of their rotation about the fixed point. This statement is validated by the circular motion of the moon that always shows its one side toward the earth.

## Case Study

The disc of the radius 0.02 m, the mass of 0.1 kg that located on the length 0.4 m from the fixed point rotates with the constant angular velocity of 5 rad/s. Determine the values of the centrifugal force and the inertial torque acting on the disc.

Solution

The value of the centrifugal force is as follows:

F = ma = mrω2 = 0,1 × 0,4 ×52 = 1,0 N

The disc turns around its center mass under the action of the inertial torque of the value: Where J is the disc mass moment of inertia about the fixed point.

## Results and Discussion

The mathematical models for the radial and angular accelerations of the rotating body about a fixed point presented in the textbooks have missed components in the analytical approach and interpretation. An analysis of the rotation of the body around the fixed point demonstrated, apart from the centrifugal force, on the body is acting the inertial torque that turns one at the process of its circular motion. The conducted analysis explains the physics of the inertial torque acting on the body rotating around the fixed point and validated by practice. The inertial torque is generated by the kinetic energy of the body’s rotation.

## Conclusion

The new analysis of the rotation of the body around a fixed point revealed the action of the inertial torque about its center mass. This inertial torque is generated by the kinetic energy of the circular motion of the object. The obtained result presents a new interpretation for the radial and angular accelerations of a rotating body. The physics mechanics enriched by the new inertial torque acting on a rotating body that presents the fundamental principle of classical mechanics. The new inertial torque should be presented in the textbooks and used by practitioners in engineering.

#### Bibliography

1. Taylor JR. “Classical Mechanics”. Sausalito CA: University Science Books (2004).
2. Gregory DR. “Classical Mechanics”. Cambridge University Press, New York (2006).
3. Hibbeler RC. “Engineering Mechanics - Statics and Dynamics”. 12th ed. Prentice Hall, Pearson, Singapore (2010).
4. Dreizler R. “Theoretical Mechanics: Theoretical Physics 1”. Springer, Boca Raton (2010).
5. Jewett J and Serway RA. “Physics for Scientists and Engineers, Cengage Learning”. 10 ed. USA, Boston (2018).
6. Smith P F and Longley WR. “Theoretical Mechanics”. Franklin Classics, Boston (2018).
7. Germain P., et al. “Theoretical and Applied Mechanics”. Elsevier (2012).
8. Nolting W. “Theoretical Physics 1: Classical Mechanics”. Springer, Cham, Switzerland (2016).
9. Aardema MD. “Analytical Dynamics”. Theory and Application. Academic/Plenum Publishers, New York (2005).
10. Thornton M. “Classical dynamics of particles and systems”. (5th ed.). Brooks/Cole (2004).
11. Baumann G. Mathematica for Theoretical Physics, Springer, New York (2003)
12. Millard B F. “Principles of Engineering Mechanics: Dynamics-The Analysis of Motion”. Springer (2006).
13. Kevin Russell., et al. “Kinematics and Dynamics of Mechanical Systems”. 2nd ed.: Implementation in MATLAB and Sim Mechanics, Taylor and Francis, Boca Raton (2019).
14. Müller-Kirsten Harald JW. “Classical Mechanics and Relativity”. World Scientific (2008).
15. Knauf, Andreas. “Mathematical Physics: Classical Mechanics”. Springer (2018).
16. https://www.scientificamerican.com/article/is-it-just-a-coincidence/

#### Citation

Citation: Ryspek Usubamatov. “Analysis of Inertial Torque Acting on a Rotating Body". Acta Scientific Computer Sciences 3.6 (2021): 09-11.

Copyright: © 2021 Ryspek Usubamatov. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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