Analysis of Newton's Second Law and Effective Mass using Energy Conservation Principles:

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803

05-06-2024

This study delves into Newton's second law of motion and the concept of effective mass in the context of classical mechanics, considering the variability of effective mass depending on the object's velocity and kinetic energy.

In classical mechanics, effective mass (mᵉᶠᶠ) embodies the mass equivalent of an object's kinetic energy in motion. While it is often comparable to or less than inertial mass (m), scenarios exist where effective mass (mᵉᶠᶠ) surpasses inertial mass (m), notably at high velocities or with significant kinetic energies. Thus, the equivalence between effective mass (mᵉᶠᶠ) and inertial mass (m) is contingent upon the specific circumstances of motion.

Considering Newton's second law (F = m⋅a), it becomes apparent that an increase in force (F) directly yields a rise in acceleration (a), showcasing a direct proportionality (F ∝ a). Furthermore, for a constant force, acceleration (a) inversely relates to the mass (m) of the object (a ∝ 1/m). Hence, alterations in effective mass (mᵉᶠᶠ) due to changes in velocity or kinetic energy reflect directly in the object's acceleration.

In the context of energy conservation principles, effective mass (mᵉᶠᶠ) corresponds to the kinetic energy (KE) of the object, delineating a scenario where mᵉᶠᶠ = KE. However, it's crucial to discern that this equivalence does not necessitate the equality of gravitational potential energy (PE) and kinetic energy (KE). Typically, PE and KE denote distinct forms of energy within the system and are seldom equal unless under specific conditions.

While the approximation that effective mass (mᵉᶠᶠ) cannot exceed inertial mass (m) aligns with classical mechanics' typical assumptions, it's imperative to acknowledge its potential deviations in certain circumstances. Particularly in systems subjected to extreme conditions or non-conservative forces, the relationship between effective mass and inertial mass may diverge from conventional expectations, emphasizing the need for meticulous analysis and consideration of the specific motion dynamics involved.

The general form of the total energy (Eᴛᴏᴛ) equation is expressed as:

Eᴛᴏᴛ = PE + KE

where Eᴛᴏᴛ is the total energy of the system, KE is the kinetic energy, and PE is the potential energy.

For an object of inertial mass m moving at velocity v and under the influence of gravitational potential energy, the total energy equation can be written as:

Eᴛᴏᴛ = (1/2)mv² + mgh

where (1/2)mv² is the kinetic energy, and mgh is the gravitational potential energy, with g being the acceleration due to gravity and h the height above a reference level.

In relativistic mechanics, the total energy equation includes rest mass energy and is expressed as:

Eᴛᴏᴛ = γmc²

where γ is the Lorentz factor {= 1/√(1-v²/c²)}, m is the rest mass of the object, and c is the speed of light.

In classical mechanics, elastic deformations occur when the atomic or molecular structure of materials rearranges under the influence of external forces. The speeds involved in these deformations are significantly less than the speed of light, assuming no nuclear reactions or changes within the materials. Piezoelectric materials, including accelerometer devices, exhibit clear empirical evidence of these occurrences when subjected to mechanical forces and gravity. Therefore, the relativistic energy-mass equivalence equation E=mc²  does not apply at speeds significantly less than the speed of light without nuclear reactions or changes within the object materials.

The concept of effective mass (mᵉᶠᶠ) represents the mass of a particle considering not only its inertial properties but also the influence of external forces, such as gravitational or electromagnetic fields, and kinetic energy. It is particularly useful in scenarios where the particle's behaviour is influenced by motion and the surrounding environment, where interactions between particles alter the apparent mass, without nuclear reactions or changes within the object materials.

The concept of effective mass (mᵉᶠᶠ) is introduced in the context of classical mechanics and kinetic energy. This idea finds support in the research paper titled "Dark energy and the structure of the Coma cluster of galaxies" by A. D. Chernin, et al. The paper explores the implications of dark energy on the foundational principles of Newtonian mechanics within galaxy clusters, investigating the behaviour of celestial entities. The findings suggest that dark energy influences the dynamics of galaxy clusters, challenging and expanding our understanding of classical mechanics. In this context, it is valid to interpret effective mass in terms of kinetic energy, particularly when considering the influence of external forces and the motion of celestial bodies within these clusters. Therefore, it is accurate to assert that the concept of effective mass, as used in the study of galaxy clusters and dark energy, is closely related to kinetic energy. 

Analysing Newton's second law of motion reveals that when the potential energy of inertial mass (m) decreases due to the application of force and corresponding acceleration, an equivalent kinetic energy is generated, which can be represented as effective mass (mᵉᶠᶠ). This means the inertial mass (m) can be viewed in terms of the object's kinetic energy (KE), which is represented as effective mass (mᵉᶠᶠ). Thus, the expression of total energy (Eᴛᴏᴛ) becomes the sum of inertial mass (m) and effective mass (mᵉᶠᶠ), expressed as:

Eᴛᴏᴛ = m + mᵉᶠᶠ

where the inertial mass (m) and effective mass (mᵉᶠᶠ) represent the potential energy (PE) of the inertial mass (m) and the kinetic energy (KE) due to the motion of the effective mass (mᵉᶠᶠ), respectively.

Effective mass (mᵉᶠᶠ)

Definitive Description:

Effective mass (mᵉᶠᶠ) is a concept in physics that represents the mass of a particle when taking into account not only its inertial properties but also the influence of external forces, such as gravitational or electromagnetic fields, as well as kinetic energy. It is particularly useful in scenarios where the behaviour of the particle is affected by its motion and the surrounding environment, such as in relativistic mechanics or within certain materials where interactions between particles alter the apparent mass.

Example 1:  1% of the Speed of Light

Consider a 10-gram object accelerating to 1% of the speed of light (approximately 2997924.58 m/s). The object's effective mass can be determined by accounting for the kinetic energy as per the respective speed, finding kinetic energy using KE = (1/2)mv² in Jules, and then determining the effective mass mᵉᶠᶠ based on the values of KE in joules.

Given Values:

• Inertial mass (m): 10 grams = 0.01 kg

• Velocity (v): 2997924.58 m/s (0.01c)

• Time (t): 10000 seconds

Calculation Steps:

Calculate Kinetic Energy:

KE = (1/2) × 0.01 kg × (2997924.58 m/s)²

KE = 4.494 × 10¹¹ J 

Calculate Effective Mass:

mᵉᶠᶠ = (2 × 4.494 × 10¹¹ J)/(2997924.58 m/s)²

mᵉᶠᶠ ≈ 14.97 kg

In this example, the effective mass (mᵉᶠᶠ) is not necessarily equal to the inertial mass (m) due to the significant velocity of 1% of the speed of light. At such speeds, the relativistic effects become non-negligible, causing the effective mass to deviate from the inertial mass. Therefore, mᵉᶠᶠ differs from 0.01 kg when considering the kinetic energy (KE) associated with the given velocity.

Example 2: 10% of the Speed of Light

Given Values:

• Inertial mass (m): 10 grams = 0.01 kg

• Velocity (v): 29979245.8 m/s (0.1c)

• Time (t): 10000 seconds

Calculate Kinetic Energy:

KE = (1/2) × 0.01 kg × (29979245.8 m/s)²

KE = 4.494 × 10¹³ J 

Calculate Effective Mass:

mᵉᶠᶠ = (2 × 4.494 × 10¹³ J)/(29979245.8 m/s)²

mᵉᶠᶠ ≈ 149.97 kg

In this example, the effective mass (mᵉᶠᶠ) is not necessarily equal to the inertial mass (m) due to the significant velocity of 10% of the speed of light.

Example 3: 50% of the Speed of Light

Given Values:

Inertial mass (m) = 10 grams = 0.01 kg

Velocity (v) = 149896229 m/s (0.5c)

Time (t) = 10000 seconds

Calculate Kinetic Energy:

KE = (1/2) × 0.01 kg × (149896229 m/s)²

KE = 1.124 × 10¹⁵ J 

Calculate Effective Mass:

mᵉᶠᶠ = (2 × 1.124 × 10¹⁵ J)/(149896229 m/s)²

mᵉᶠᶠ ≈ 374.92 kg

In this example, the effective mass (mᵉᶠᶠ) is not necessarily equal to the inertial mass (m) due to the significant velocity of 50% of the speed of light.

Summary of Effective mass:

1. 1% of the Speed of Light:

mᵉᶠᶠ ≈ 14.97 kg

2. 10% of the Speed of Light:

mᵉᶠᶠ ≈ 149.97 kg

3. 50% of the Speed of Light:

mᵉᶠᶠ ≈374.91 kg

These examples illustrate the variability of effective mass with increasing velocities and highlight the importance of considering relativistic effects in such scenarios.

Interpretation:

The effective mass concept helps us understand that the mass of an object can appear different when influenced by external forces or when moving at significant velocities. In these examples, the effective mass differs from the inertial mass due to relativistic effects at higher velocities. This deviation illustrates the dynamic nature of mass in various physical contexts and emphasizes the need to account for relativistic effects in accurate predictions and analyses.

Is it possible to exceed the speed of light by using a black hole? Would this be considered faster-than-light travel?

Nothing can travel faster than light in a gravitationally bound system. Gravity won't allow that. In 1899, Max Planck proposed fundamental natural units for length, mass, time, and energy. He used dimensional analysis to derive these units, including Planck length (ℓP) and the speed of light (c). Planck length is equal to one Planck length per Planck time (ℓP/tP=c), which describes the speed at which photons travel.

However, beyond the gravitational influence of galaxies or clusters, anti-gravity caused by dark energy can occur, allowing entire galaxies or clusters to recede from each other at effective speeds greater than the speed of light. This is due to the repulsive force of dark energy acting on massive bodies, rather than their motion through space. This concept is crucial in understanding the true abstract nature of space and time rather than natural space-time.

The Planck length is expected to be the shortest measurable distance. Any attempt to investigate the possible existence of shorter distances by performing higher-energy collisions would inevitably result in black hole production, according to some theories of quantum gravity.

Relevant URL https://qr.ae/psJt9h

Comparative Analysis of Length Deformation in Classical and Relativistic Mechanics: Part-2

Soumendra Nath Thakur                             Link URL of the research

20-05-2024

Abstract:

This study, serving as Part-2 of the research titled "Comparative Analysis of Length Deformation in Classical and Relativistic Mechanics,Comparative Analysis of Length Deformation in Classical and Relativistic Mechanics," investigates the behaviour of matter within gravitationally bound systems. Through meticulous examination of projected length alterations, the research highlights differences between classical and relativistic mechanics frameworks, emphasizing the necessity of considering relativistic effects beyond velocity alone. Additionally, the study underscores the crucial role of gravitational effects on the effective mass of moving objects, which emerges as a critical factor in predicting length deformation across scientific disciplines. The incomplete treatment of relativistic effects within Relativistic Mechanics, including acceleration and material stiffness, emphasizes the importance of comprehensively understanding gravitational influences on effective mass. This is evident in gravitational equations, where the gravitational force depends not only on the object's mass but also on its effective mass, influenced by kinetic energy. Thus, incorporating the gravitational effect on effective mass enhances the understanding of length deformation phenomena within gravitationally bound systems, enriching scientific discourse.

Keywords: Length Deformation, Classical Mechanics, Relativistic Mechanics, Gravitational Effects, Effective Mass,

Comment: The previous research titled "Comparative Analysis of Length Deformation in Classical and Relativistic Mechanics" offers valuable insights into the differences between classical and relativistic predictions of length deformation. However, a Part 2 of this research, titled "Comparative Analysis of Length Deformation in Classical and Relativistic Mechanics: Part-2," could further enhance our understanding in several ways. It could delve deeper into relativistic dynamics, explore alternative frameworks, validate theoretical predictions through experiments, extend the analysis to different scenarios, integrate quantum mechanics, and discuss broader implications and applications. By addressing these aspects, Part 2 could provide a more comprehensive and nuanced perspective on length deformation phenomena in extreme velocity scenarios.

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803

Tagore’s Electronic Lab, W.B. India

Emails: postmasterenator@gmail.com

postmasterenator@telitnetwork.in

Declarations:

Funding: No specific funding was received for this work,

Potential competing interests: No potential competing interests to declare.

Introduction:

Understanding the behaviour of matter under extreme conditions, particularly at high velocities, is a fundamental pursuit in physics. Classical and Relativistic Mechanics offer indispensable frameworks for comprehending the intricate dynamics involved in such scenarios. This research serves as a continuation of the investigation initiated in the previous study titled "Comparative Analysis of Length Deformation in Classical and Relativistic Mechanics." In this Part-2, our focus remains on exploring the phenomenon of length deformation within gravitationally bound systems.

The quest for knowledge in this domain necessitates a meticulous examination of predicted length changes, thereby illuminating the disparities between classical and relativistic mechanics frameworks. While classical mechanics provides a robust foundation rooted in principles like Hooke's Law, Relativistic Mechanics introduces nuanced considerations, particularly concerning the interplay of velocity and gravitational effects.

Moreover, the research underscores the pivotal role of gravitational effects on the effective mass of moving objects. The effective mass, modulated by kinetic energy, emerges as a critical factor in forecasting length deformation across scientific disciplines. This emphasis on gravitational effects on effective mass is particularly relevant given the complexities inherent in understanding the behaviour of matter within gravitationally bound systems.

This study delves into the nuanced interplay between classical and relativistic mechanics, particularly emphasizing the importance of considering relativistic effects beyond velocity alone. By scrutinizing the implications of acceleration dynamics and the incomplete treatment of certain factors in Relativistic Mechanics, we aim to deepen our understanding of length deformation in high-speed scenarios.

Through rigorous analysis and comparison of derived length changes, this research endeavours to elucidate the divergent predictions of classical and relativistic frameworks. Furthermore, we seek to underscore the critical role of gravitational effects on the effective mass of moving objects, highlighting its significance in accurately predicting length deformation across scientific disciplines.

In essence, this research aims to contribute to the ongoing dialogue surrounding the behaviour of matter under extreme velocities, thereby enriching our comprehension of the transition between classical and relativistic regimes. By shedding light on the nuanced considerations within each framework, we endeavour to advance our understanding of length deformation phenomena within gravitationally bound systems.

Methodology:

1. Application Setup:

• Compare length deformation predictions in both classical and relativistic mechanics frameworks.

• Use a 10-gram object as the subject of analysis, ensuring consistency in mass between classical and relativistic calculations.

• Employ a mechanism capable of applying a known force to the object and measuring the resulting displacement accurately.

2. Classical Mechanics Application:

• Apply a known force to the object using the designed mechanism.

• Measure the resulting displacement of the object.

• Calculate the change in length using Hooke's Law and the formula ΔL = F/k, where k is the spring constant derived from the applied force and the object's displacement.

3. Relativistic Mechanics Application:

• Repeat the force application process with the same 10-gram object.

• Apply the resulting displacement in the Lorentz Factor to account for relativistic effects.

• Calculate the change in length using the Lorentz contraction formula L = L₀√(1-v²/c²), where L₀ is the proper length, v is the velocity of the object, and c is the speed of light.

4. Data Collection and Analysis:

• Record the derived length changes obtained from both classical and relativistic mechanics applications.

• Compare the length deformation predictions between the two methodologies.

• Evaluate the discrepancy between classical and relativistic predictions, considering factors such as material stiffness, proportionality constant and velocity-dependent contraction.

• Analyse the impact of gravitational effects on effective mass and its role in length deformation predictions.

5. Discussion and Interpretation:

• Discuss the findings in the context of classical and relativistic mechanics theories.

• Analyse the significance of observed differences in length deformation predictions.

• Explore the applicability and limitations of the Lorentz Factor in describing length deformations under high-speed conditions.

• Consider the broader implications of the study's results for understanding matter behaviour at extreme velocities.

6. Conclusion and Future Directions:

• Summarize the key findings and insights gained from the study.

• Identify areas for further research, including potential refinements to the experimental setup or theoretical frameworks.

• Discuss potential applications of the study's findings in fields such as astrophysics, particle physics, and engineering.

Mathematical Presentation:

Example Calculation:

To illustrate the application of the methodology, we calculate the effective mass mᵉᶠᶠ  and corresponding length deformation in classical mechanics:

1. Given Values:

• m (inertial mass): 10 grams = 0.01 kg

• v (velocity): 2997924.58 m/s = 0.01c

• t (time): 10000 seconds

• ΔL (length change): 0.1 millimetres = 0.0001 meters

2. Calculate Acceleration:

a = v/t = (2997924.58 m/s) / (10000 s)

= 299.792458 m/s²

In the given equation:

• v is the initial velocity of the object, which is 2997924.58 meters per second (approximately the speed of light).

• t is the time interval over which the velocity change occurs, which is 10000 seconds.

• a is the resulting acceleration, which is 299.792458 meters per second squared.

This equation demonstrates how to calculate acceleration by dividing the change in velocity (v) by the time interval (t). In this specific example, it calculates the acceleration of an object moving at approximately 1% of the speed of light over a time interval of 10000 seconds. The resulting acceleration value is approximately 299.792458 meters per second squared.

3. Calculate Force:

F = m⋅a

F = 0.01 kg × 299.792458 m/s²

F = 2.99792458 N

In the given example:

• m is the mass of the object, which is 0.01 kilograms.

• a is the acceleration of the object, which is 299.792458 meters per second squared.

• F is the resulting force exerted on the object, which is 2.99792458 Newton.

This equation demonstrates how to calculate the force acting on an object when its mass and acceleration are known. In this specific example, it calculates the force exerted on an object with a mass of 0.01 kilograms experiencing an acceleration of 299.792458 meters per second squared. The resulting force is approximately 2.99792458 Newton.

4. Explanation:

Based on the force and acceleration provided, mᵉᶠᶠ equals the inertial mass m. This suggests mᵉᶠᶠ represents the dynamic response to the applied force, consistent with Newton's second law.

Total Energy Equation:

Eᴛᴏᴛ = PE + KE = m + mᵉᶠᶠ

In the given example:

• Eᴛᴏᴛ is the total energy of the object.

• PE is the potential energy of the object.

• KE is the kinetic energy of the object.

• m represents the inertial mass of the object.

• mᵉᶠᶠ represents the effective mass due to kinetic energy.

Here, m is the rest mass (0.01 kg) and mᵉᶠᶠ is the effective mass due to kinetic energy (0.01 kg).

The equation relates the total energy of an object to its potential energy and kinetic energy. It suggests that the total energy of the object is the sum of its inertial mass m and the effective mass mᵉᶠᶠ due to kinetic energy. This equation accounts for both the rest mass of the object and the additional mass gained due to its motion, represented by the effective mass mᵉᶠᶠ.

5. Effective Mass Calculation:

mᵉᶠᶠ =F/a

mᵉᶠᶠ = (2.99792458 N)/(299.792458 m/s²)

mᵉᶠᶠ = 0.01kg

• mᵉᶠᶠ represents the effective mass due to kinetic energy.

6. Conclusion:

Given the values and steps, the effective mass mᵉᶠᶠ calculated:

mᵉᶠᶠ = 0.01 kg

This is consistent with classical mechanics:

• Inertial mass m: 0.01 kg

• Effective mass mᵉᶠᶠ: 0.01 kg

Thus, the force of 2.99792458 N corresponds to the effective mass mᵉᶠᶠ = 0.01 kg due to the given acceleration. The classical mechanics framework holds without relativistic effects, aligning the calculations with Newtonian principles

7. Gravitational Force Calculation:

Given the mass of Earth m₁, the gravitational force equation considering effective mass is:

F = G·{m₁·(m + mᵉᶠᶠ)}/r²

In the equation:

• F represents the gravitational force between two objects.

• G is the universal gravitational constant, approximately 6.674 × 10⁻¹¹ N⋅m²/kg² representing the strength of the gravitational force.

• m₁ is the mass of one of the objects involved in the interaction, here Earth, 5.972 × 10²⁴ kg.

• m is the inertial mass of the object, 0.01 kg

• mᵉᶠᶠ is the effective mass due to kinetic energy, 0.01 kg.

• r is the distance between the centres of the two objects, 1 metre.

Substitute the values:

F=6.674×10⁻¹¹·{(5.972×10²⁴)·(0.01+ 0.01)}/1²

F ≈ 7.97×10¹² N

Substitute the values:

F=6.674×10⁻¹¹·{(5.972×10²⁴)·(0.01+ 0.01)}/1²

F ≈ 7.97 × 10¹² N

This equation evaluates the gravitational force F acting between two objects. In this specific instance, it determines the gravitational interaction between one object with a mass equivalent to that of the Earth (denoted as m in kilograms) and another object with a total mass of 0.02 kilograms, comprising both its inertial mass m and its effective mass mᵉᶠᶠ. The separation between these objects is fixed at 1 meter. The resultant gravitational force approximates to 7.97 × 10¹² Newton.

This formulation takes into account both the inertial mass and the additional effective mass attributable to kinetic energy within the gravitational interaction. Thus, it yields a force arising from the gravitational influence when interacting with the Earth's mass at a distance of 1 meter. This approach effectively integrates kinetic energy contributions into mass-like effects within classical mechanics, as confirmed by the applied force and the derived effective mass. By incorporating the effective mass originating from kinetic energy into the gravitational force equation, the calculations maintain alignment with the fundamental principles of Newtonian mechanics.

By adhering to this systematic methodology, researchers can methodically explore and compare predictions of length deformation in classical and relativistic mechanics, thereby enhancing our comprehension of material behaviour under extreme circumstances.

Consequence of Gravitational Force in Upward Motion in Space:

In the scenario where the motion is directed vertically upward, away from the Earth, the consequence of the gravitational force is a gradual decrease in acceleration as the object moves farther from the Earth's surface. As the object moves away from the gravitational influence of the Earth, the force of gravity diminishes in accordance with the inverse square law, resulting in a reduction in the object's acceleration. Eventually, at a significant distance from the Earth, the gravitational force becomes negligible, and the object's motion may become influenced by other celestial bodies or external forces. This phenomenon highlights the dynamic nature of gravitational interactions in space and underscores the importance of considering gravitational effects on objects moving away from planetary surfaces.

Discussion:

The research study delves into the behaviour of matter within gravitationally bound systems, aiming to elucidate the discrepancies between classical and relativistic mechanics frameworks regarding length deformation. This discussion provides an analysis of the research paper, covering key aspects such as the methodology employed, findings, and implications.

Methodology:

The methodology outlined in the research paper establishes a systematic approach to compare length deformation predictions in classical and relativistic mechanics frameworks. By employing a consistent mass for analysis and utilizing appropriate equations from classical and relativistic mechanics, the study ensures a fair comparison. The inclusion of both classical and relativistic mechanics applications allows for a comprehensive examination of length deformation phenomena under different theoretical frameworks.

Findings and Interpretation:

The research findings underscore the importance of considering relativistic effects, particularly in scenarios involving high velocities and gravitational interactions. By comparing length deformation predictions derived from classical and relativistic mechanics, the study highlights significant disparities, emphasizing the necessity of accounting for relativistic corrections beyond velocity alone. Furthermore, the analysis of effective mass due to kinetic energy sheds light on the nuanced dynamics underlying length deformation in gravitationally bound systems.

Implications:

The implications of the research extend beyond theoretical physics, encompassing diverse scientific disciplines. By elucidating the role of gravitational effects on effective mass and its impact on length deformation predictions, the study offers insights applicable to fields such as astrophysics, particle physics, and engineering. Moreover, the research underscores the dynamic nature of gravitational interactions in space, emphasizing the need to consider gravitational effects on objects moving away from planetary surfaces.

Conclusion and Future Directions:

In conclusion, "Comparative Analysis of Length Deformation in Classical and Relativistic Mechanics: Part-2" contributes to advancing our understanding of matter behaviour under extreme conditions. Moving forward, future research could explore additional factors influencing length deformation predictions, such as non-uniform gravitational fields or relativistic corrections beyond the scope of this study. Furthermore, the application of findings from this research in practical contexts, such as spacecraft design or particle accelerator technologies, holds promise for driving technological innovation and scientific discovery.

Overall, the research paper provides a valuable contribution to scientific discourse, fostering dialogue and further exploration of length deformation phenomena within gravitationally bound systems.

Conclusion:

In this study, we embarked on a comprehensive exploration of length deformation phenomena within gravitationally bound systems, comparing predictions derived from classical and relativistic mechanics frameworks. Through meticulous analysis and rigorous methodology, we uncovered significant disparities in length deformation predictions, emphasizing the necessity of considering relativistic corrections and gravitational effects beyond velocity alone.

Our findings underscore the dynamic interplay between classical and relativistic mechanics, highlighting the limitations of classical approaches in predicting length alterations under extreme conditions. The analysis of effective mass due to kinetic energy provided valuable insights into the nuanced dynamics underlying length deformation in high-speed scenarios, enriching our understanding of material behaviour within gravitationally bound systems.

Furthermore, the implications of our research extend beyond theoretical physics, encompassing diverse scientific disciplines such as astrophysics, particle physics, and engineering. By elucidating the role of gravitational effects on effective mass and their impact on length deformation predictions, our study contributes to advancing scientific discourse and fostering technological innovation.

In conclusion, "Comparative Analysis of Length Deformation in Classical and Relativistic Mechanics: Part-2" enriches our understanding of length deformation phenomena within gravitationally bound systems. By shedding light on the dynamic interplay between classical and relativistic mechanics frameworks, our research paves the way for further exploration and technological advancements in fields ranging from space exploration to particle accelerator technologies.

References:

[1] Chernin, A. D., Бисноватый-коган, Г. С., Teerikorpi, P., Valtonen, M. J., Byrd, G. G., & Merafina, M. (2013a). Dark energy and the structure of the Coma cluster of galaxies. Astronomy and Astrophysics, 553, A101. https://doi.org/10.1051/0004-6361/201220781

[2] Thakur, S. N. (2023p). Decoding Time Dynamics: The Crucial Role of Phase Shift Measurement amidst Relativistic & Non-Relativistic Influences, Qeios, https://doi.org/10.32388/mrwnvv

[3] Thakur, S. N. (2024a). Decoding nuances: relativistic mass as relativistic energy, Lorentz’s transformations, and Mass-Energy. . . EasyChair, 11777, https://doi.org/10.13140/RG.2.2.22913.02403

[4] Thakur, S. N. (2024f). Direct Influence of Gravitational Field on Object Motion invalidates Spacetime Distortion. Qeios, BFMIAU. https://doi.org/10.32388/bfmiau

[5] Thakur, S. N. (2024i). Introducing Effective Mass for Relativistic Mass in Mass Transformation in Special Relativity and. . . ResearchGate https://doi.org/10.13140/RG.2.2.34253.20962

[6] Thakur, S. N. (2024k). Re-examining Time Dilation through the Lens of Entropy: ResearchGate, [11] https://doi.org/10.13140/RG.2.2.36407.70568

[7] Thakur, S. N. (2024m). Re-examining time dilation through the lens of entropy, Qeios, https://doi.org/10.32388/xbuwvd

[8] Thakur, S. N. (2024n). Standardization of Clock Time: Ensuring Consistency with Universal Standard Time. EasyChair, 12297, https://doi.org/10.13140/RG.2.2.18568.80640

[9] Thakur, S. N., & Bhattacharjee, D. (2023j). Phase shift and infinitesimal wave energy loss equations. Journal of Physical Chemistry & Biophysics, 13(6), 1000365 https://www.longdom.org/open-access/phase-shift-and-infinitesimal-wave-energy-loss-equations-104719.html

[10] Morin, D. (2008). Introduction to Classical Mechanics: With Problems and Solutions.

[11] Misner, C. W., Thorne, K. S., & Wheeler, J. A. (1973). Gravitation

[12] Maoz, D. (2007), Astrophysics in a Nutshell

#classicalmechanics #effectivemass #gravitationaleffects #LengthDeformation #RelativisticMechanics

Comparative Analysis of Length Deformation in Classical and Relativistic Mechanics

Soumendra Nath Thakur                               URL Of the Research

Date: 18-05-2024

Abstract

This study presents a comparative analysis of length deformation in Classical and Relativistic Mechanics, specifically investigating 10-gram objects accelerating to 1% of the speed of light. By employing Hooke's Law in Classical Mechanics and the Relativistic Lorentz Factor, the research explores the implications of acceleration dynamics and the limitations inherent in Relativistic Mechanics. The results reveal significant differences in predicted length changes between the two frameworks, emphasizing the necessity of considering relativistic effects beyond velocity alone. This study underscores the critical importance of addressing the incomplete treatment of acceleration dynamics in Relativistic Mechanics to achieve a more accurate depiction of length deformation in high-speed scenarios.

Keywords: Length Deformation, Classical Mechanics, Relativistic Mechanics, Hooke's Law, Lorentz Factor, Acceleration,

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803

Tagore’s Electronic Lab, W.B. India

Emails: postmasterenator@gmail.com

postmasterenator@telitnetwork.in

Declarations:

Funding: No specific funding was received for this work,

Potential competing interests: No potential competing interests to declare.

Introduction

In the study of objects' behaviour under extreme velocities, Classical and Relativistic Mechanics offer indispensable frameworks. This research explores the phenomenon of length deformation experienced by 10-gram objects accelerating to 1% of the speed of light (v = 2997924.58 m/s = 0.01c) over a duration of 10,000 seconds. By incorporating principles from Classical Mechanics, notably Hooke's Law, and Relativistic Mechanics, utilizing the Lorentz Factor, we scrutinize the implications of acceleration and the oversight of certain factors in Relativistic Mechanics. The study aims to elucidate the differences in derived length changes between the two methodologies and to discern the extent to which relativistic effects influence observed deformations.

Through rigorous analysis of the derived length changes using both Classical and Relativistic perspectives, this research seeks to illuminate the nuanced interplay between acceleration, velocity, and relativistic effects. The comparison between deformation results obtained from Classical Mechanics, which integrates factors like material stiffness and proportionality constant, and Relativistic Mechanics, which accounts for velocity-dependent contraction, promises to elucidate the divergent predictions of these theoretical frameworks.

By meticulously examining and calculating the degree of discrepancy between the length deformations predicted by Classical and Relativistic Mechanics, this study aims to deepen our understanding of the transition between Classical and Relativistic regimes. Additionally, by investigating the applicability and limitations of the Lorentz Factor, particularly concerning the minimum speed at which its effects become significant, this research seeks to contribute to the ongoing dialogue surrounding the behaviour of matter at extreme velocities.

In essence, this research endeavour aims to provide insights into the intricate interplay between Classical and Relativistic Mechanics in describing length deformations under high-speed conditions, contributing to our understanding of matter's behaviour at extreme velocities.

Mechanism

To conduct the study comparing length deformation in Classical and Relativistic Mechanics, we propose the following mechanism:

1. Classical Mechanics Application:

• Apply the known force to the object using the designed mechanism.

• Apply the resulting displacement of the object.

• Calculate the change in length using Hooke's Law and the formula ΔL = F/k, where k is the spring constant derived from the applied force and the object's displacement.

2. Relativistic Mechanics Application:

• Repeat the force application process with the same 10-gram object.

• Apply the resulting displacement in the Lorentz Factor to account for relativistic effects.

• Calculate the change in length using the Lorentz contraction formula L =L₀√(1-v²/c²), where L₀ is the proper length, v is the velocity of the object, and c is the speed of light.

3. Data Analysis:

• Compare the derived length changes obtained from Classical and Relativistic mechanics applications.

• Evaluate the discrepancy between the two methodologies and assess the impact of factors such as material stiffness, proportionality constant and velocity-dependent contraction.

• Consider the implications of inevitable acceleration and the oversight of certain factors in Relativistic Mechanics on the observed length deformations.

4. Discussion and Interpretation:

• Discuss the findings in the context of Classical and Relativistic Mechanics theories.

• Analyse the significance of the observed differences in length deformation predictions.

• Explore the applicability and limitations of the Lorentz Factor in describing length deformations under high-speed conditions.

• Consider the broader implications of the study's results for our understanding of matter behaviour at extreme velocities.

5. Conclusion and Future Directions:

• Summarize the key findings and insights gained from the study.

• Identify areas for further research and experimentation, including potential refinements to the experimental setup or theoretical frameworks.

• Discuss the potential applications of the study's findings in fields such as astrophysics, particle physics, and engineering.

By following this proposed mechanism, researchers can systematically investigate and compare length deformation predictions in Classical and Relativistic Mechanics, advancing our understanding of matter behaviour under extreme conditions.

Mathematical Presentation

1. Relativistic Derivation of Length Contraction with Lorentz Factor

Lorentz Factor (γ) Derivation:

The Lorentz factor is defined as: γ= 1/√(1-v²/c²)

For an object moving at 1% of the speed of light:

v = 0.01c

Plugging into Lorentz factor equation:

γ = √{1-(0.01c/c)²} = 1/√0.9999 ≈1.00005

Length Contraction Calculation:

The formula for length contraction is: 

L = L₀√(1-v²/c²)

Given:

v = 0.01c 

L₀ =1 metre

Substituting the values:

L = 1 × √{1−(0.01)²} ≈ 0.99995 meters

The contracted length:

ΔL = 1 − 0.99995 = 0.00005 metres = 0.05 millimetres

Summary of Relativistic Contraction:

• At 1% of the speed of light, length contraction is minimal.

• The contraction factor is approximately 0.99995, leading to a length change of 0.05 mm for a 1-meter object.

2. Classical Derivation of Length Change with Hooke's Law

Hooke's Law:

F = kΔL

Where:

F is the applied force.

k is the spring constant.

ΔL is the displacement or change in length.

Given:

m = 10 grams =0.01 kg

v = 2997924.58 m/s = 0.01c

t = 10000 seconds

Calculate Acceleration:

a = v/t = (2997924.58 m/s) / (10000 s) = 299.792458 m/s² 

 

Force Calculation:

Using Newton's second law:

F = m⋅a

F = 0.01 kg × 299.792458 m/s²  =2.99792458 N

Determine Spring Constant (k):

Assuming a known displacement ΔL=0.0001 m:

k = F/ΔL = 2.99792458 N / 0.0001 m = 29979.2458 N/m

Calculate Length Change:

Using Hooke's Law:

ΔL = F/k = (2.99792458 N) / (29979.2458 N/m) = 0.0001 m = 0.1 millimetres

Summary of Classical Deformation:

• For a force of 2.9979 N applied to a 10-gram object, the length change is 0.1 mm. This calculation assumes the proportionality constant k derived from the applied force and displacement.

Acceleration and Length Changes Between Rest Frames and Separation:

• In Classical Mechanics, acceleration is accounted for directly using F = m⋅a.

• In Relativistic Mechanics, acceleration is less straightforward due to the dependence of mass on velocity.

Velocity Changes After Attaining Desired Velocity:

• Classical Mechanics considers the force required to maintain and change velocity, incorporating acceleration.

• Relativistic Mechanics uses the Lorentz factor, which only considers the object once it is in motion, not accounting for the force and acceleration required to reach that velocity.

Comparison and Conclusions:

Classical vs. Relativistic Mechanics:

• Classical Mechanics provides a straightforward calculation of length change based on Hooke's Law, accounting for force, stiffness, and acceleration.

• Relativistic Mechanics focuses on the contracted length once an object reaches a significant fraction of the speed of light, using the Lorentz factor.

Observations:

• At 1% of the speed of light, relativistic effects are minimal (Lorentz factor γ ≈ 1.00005).

• The classical calculation predicts a greater length change (0.1 mm) compared to the relativistic prediction (0.05 mm).

Implications:

• The study highlights the differences in predicting length changes under extreme velocities.

• Classical Mechanics considers inevitable acceleration, material stiffness, and force application.

• Relativistic Mechanics primarily focuses on the contraction during uniform motion.

This detailed mathematical presentation showcases the derivations and comparisons of length deformation predictions in Classical and Relativistic Mechanics, providing insights into their respective frameworks and relativistic limitations.

Discussion

The study presents a comparative analysis of length deformation in classical mechanics and relativistic mechanics for a 10-gram object accelerating to 1% of the speed of light over 10,000 seconds. Classical mechanics utilizes Hooke's Law to determine the deformation, while relativistic mechanics employs the Lorentz factor to calculate length contraction.

Comparison of Results:

1. Classical Mechanics and Hooke's Law:

In classical mechanics, Hooke's Law states that the deformation (ΔL) of an object is directly proportional to the applied force (F) and inversely proportional to the spring constant (k). For the 10-gram object, the force calculated to achieve the given acceleration is approximately 2.9979 N. Using Hooke's Law, the change in length (ΔL) is found to be 0.0001 meters (0.1 mm).

2. Relativistic Mechanics and Lorentz Factor:

The Lorentz factor (γ) accounts for relativistic effects that become significant at high velocities, close to the speed of light. At 1% of the speed of light (v=0.01c), the Lorentz factor is approximately 1.00005, indicating negligible relativistic effects. The length contraction calculated using the Lorentz factor is 0.00005 meters (0.05 mm).

The results indicate that the classical derivation using Hooke's Law predicts a greater length change (0.1 mm) compared to the relativistic derivation using the Lorentz factor (0.05 mm). This difference highlights the varying considerations of each approach.

Implications of Inevitable Acceleration:

The study emphasizes that classical mechanics takes into account the inevitable acceleration component when calculating deformation. In contrast, the Lorentz factor primarily considers the object’s velocity relative to the speed of light and does not explicitly include the effects of acceleration when transitioning between rest frames or while changing velocity.

1. Acceleration from Rest Frames:

When the object accelerates from a rest frame, classical mechanics provides a straightforward approach by considering the applied force and resulting deformation. This aligns with Newton's second law, where force is the product of mass and acceleration.

2. Velocity Changes After Attaining Desired Velocity:

After reaching the desired velocity (v=0.01c), any further changes in velocity would still involve acceleration. Classical mechanics accounts for this by continuously applying Newton's second law. In contrast, relativistic mechanics focuses on the velocity and its effects on length contraction, often overlooking the detailed impact of ongoing acceleration.

Significance of the Lorentz Factor:

The Lorentz factor is applicable at all speeds, but its effects become significant only at velocities approaching a substantial fraction of the speed of light. At 1% of the speed of light, the Lorentz factor is very close to 1, indicating minimal relativistic effects. Because the Lorentz factor ignores relativistic effects during accelerations, for this very reason, the length contraction derived from the Lorentz factor is smaller than the deformation predicted by Hooke's Law in the classical approach.

Practical Applications and Considerations:

1. Material Stiffness and Proportionality Constant:

In classical mechanics, the proportionality constant (k) or the stiffness of the material plays a crucial role in determining deformation. This aspect is crucial for practical applications in engineering and materials science, where understanding material behaviour under different forces is essential.

2. Relativistic Considerations in High-Speed Contexts:

While the relativistic effects are negligible at low speeds, they become crucial in high-speed contexts such as particle physics and astrophysics. Understanding these effects is essential for accurate modelling and prediction of phenomena at relativistic speeds.

Limitations and Future Directions:

Our study unveils inherent constraints within relativistic methodologies, particularly concerning their treatment of acceleration dynamics. While our analysis exposes a notable discrepancy between classical and relativistic predictions, with classical mechanics foreseeing a greater length change (0.1 mm) compared to the relativistic forecast (0.05 mm), it is crucial to recognize the foundational reason behind this difference. The Lorentz factor, integral to relativistic calculations, overlooks relativistic effects during accelerations, resulting in an underestimation of length contraction relative to the classical model employing Hooke's Law. This oversight underscores the challenge faced by relativistic mechanics in fully integrating classical acceleration dynamics, even within scenarios of high-speed motion scrutinized in our investigation.

The underestimation of changes in an object's state by relativistic mechanics due to its incomplete consideration of acceleration dynamics necessitates focused efforts in future research. Addressing this discrepancy calls for a refinement of relativistic frameworks to achieve a more comprehensive understanding of length deformation phenomena under extreme velocities.

In conclusion, our study illuminates the contrasting predictions of length deformation between classical and relativistic mechanics for a 10-gram object accelerating to 1% of the speed of light. Classical mechanics, employing Hooke's Law, anticipates a greater deformation compared to the relativistic length contraction derived from the Lorentz factor. These findings underscore the critical importance of acknowledging and reconciling the underlying reasons behind such differences.

Recognizing the underestimation of changes in an object's state by relativistic mechanics due to its incomplete consideration of acceleration dynamics prompts a call for action in future research endeavours. By bridging the gap between classical and relativistic approaches, we can pave the way for a unified and more accurate depiction of physical phenomena in high-speed contexts.

Conclusion

This study undertakes a comparative analysis of length deformation in classical and relativistic mechanics for a 10-gram object accelerating to 1% of the speed of light over a time span of 10,000 seconds. By employing Hooke's Law in classical mechanics and the Lorentz factor in relativistic mechanics, the study provides insights into the differences in predicted deformations and highlights areas for future research.

1. Classical Mechanics with Hooke's Law:

Using Hooke's Law, the study finds that the deformation (ΔL) is 0.0001 meters (0.1 mm) for the given force of 2.9979 N. This approach accounts for material stiffness and the applied force, highlighting how classical mechanics considers both acceleration and the resulting deformation.

2. Relativistic Mechanics with Lorentz Factor:

The relativistic approach, utilizing the Lorentz factor, calculates a length contraction of 0.00005 meters (0.05 mm). This minimal contraction reflects the relatively insignificant relativistic effects at 1% of the speed of light.

3. Comparison and Implications:

The classical derivation predicts a greater deformation compared to the relativistic approach. This discrepancy illustrates the different considerations and assumptions in each framework. Classical mechanics includes the impact of acceleration and material properties, while relativistic mechanics focuses on velocity and its effects on length contraction.

4. Significance of Acceleration:

The study emphasizes the role of acceleration in classical mechanics, both from rest frames and during changes in velocity after attaining a desired speed. Relativistic mechanics, on the other hand, often overlooks these acceleration dynamics, which can lead to underestimations of deformation.

5. Practical and Theoretical Insights:

For practical applications, especially in engineering and material sciences, understanding the limitations and strengths of both classical and relativistic mechanics is essential. While classical mechanics provides a robust framework for low-speed scenarios, relativistic mechanics becomes crucial at higher velocities approaching the speed of light.

6. Limitations and Future Directions:

This study unveils inherent constraints within relativistic methodologies, particularly concerning their treatment of acceleration dynamics. The underestimation of changes in an object's state by relativistic mechanics due to its incomplete consideration of acceleration dynamics necessitates focused efforts in future research. Addressing this discrepancy calls for a refinement of relativistic frameworks to achieve a more comprehensive understanding of length deformation phenomena under extreme velocities.

In conclusion, this study underscores the importance of selecting the appropriate mechanical framework based on the specific conditions and requirements of the scenario. Both classical and relativistic mechanics offer valuable insights, and their combined understanding is crucial for advancing our comprehension of motion and deformation at varying speeds.

References

[1] Carroll, S. Spacetime and Geometry: An Introduction to General Relativity.

[2] Einstein, A. Relativity: The Special and General Theory.

[3] Goldstein, H., Poole, C. P., & Safko, J. L. Classical Mechanics.

[5] Morin, D. Introduction to Classical Mechanics: With Problems and Solutions.

[6] Thakur, S. N. (2023p). Decoding Time Dynamics: The Crucial Role of Phase Shift Measurement amidst Relativistic & Non-Relativistic Influences. Qeios, MRWNVV. https://doi.org/10.32388/mrwnvv

[7] Thakur, S. N. (2024a). Decoding nuances: relativistic mass as relativistic energy, Lorentz’s transformations, and Mass-Energy. . . EasyChair, 11777. https://doi.org/10.13140/RG.2.2.22913.02403

[8] Thakur, S. N. (2024f). Direct Influence of Gravitational Field on Object Motion invalidates Spacetime Distortion. Qeios, BFMIAU. https://doi.org/10.32388/bfmiau

[9] Thakur, S. N. (2024i). Introducing Effective Mass for Relativistic Mass in Mass Transformation in Special Relativity and. . . ResearchGate. https://doi.org/10.13140/RG.2.2.34253.20962

[10] Thakur, S. N. (2024k). Re-examining Time Dilation through the Lens of Entropy: ResearchGate. [11] https://doi.org/10.13140/RG.2.2.36407.70568

[12] Thakur, S. N. (2024m). Re-examining time dilation through the lens of entropy. Qeios. https://doi.org/10.32388/xbuwvd

[13] Thakur, S. N. (2024n). Standardization of Clock Time: Ensuring Consistency with Universal Standard Time. EasyChair, 12297. https://doi.org/10.13140/RG.2.2.18568.80640

[14] Thakur, S. N., & Bhattacharjee, D. (2023j). Phase shift and infinitesimal wave energy loss equations. Journal of Physical Chemistry & Biophysics, 13(6), 1000365. https://www.longdom.org/open-access/phase-shift-and-infinitesimal-wave-energy-loss-equations-104719.html

[15] Tipler, P. A., & Llewellyn, R. Modern Physics.

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