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.
