Section 44–1 Heat engines; the first law

Gough-Joule effect / Historical development / First Law

 

In this section, Feynman discusses the Gough-Joule effect, the historical development of heat engines, and the First Law of Thermodynamics. “Rubber Bands, Historical Development, and the First Law” is a possible alternative title because it captures the section’s three core themes. By explicitly naming rubber bands, the title highlights the section’s concrete, experimentally accessible opening example. The phrase “Historical Development” reflects the evolution of thermodynamic ideas, including both the key insights and the limitations of Carnot’s theory—though Feynman’s historical remarks serve more as pedagogical asides rather than a rigorous history. Finally, referencing “The First Law” clearly identifies the central theoretical principle established in the text. Although longer than Feynman’s original title, “Heat Engines; The First Law,” it represents the section’s pedagogical progression: from an interesting phenomenon, through historical context, to a fundamental law of physics.

 

1. Gough-Joule effect

“So the real reason why a rubber band tends to contract is this: when one pulls it out, the chains are lengthwise, and the thermal agitations of the molecules on the sides of the chains tend to kink the chains up, and make them shorten. One can then appreciate that if the chains are held stretched and the temperature is increased, so that the vigor of the bombardment on the sides of the chains is also increased, the chains tend to pull in, and they are able to pull a stronger weight when heated. If, after being stretched for a time, a rubber band is allowed to relax, each chain becomes soft, and the molecules striking it lose energy as they pound into the relaxing chain. So the temperature falls (Feynman et al, 1963).”

 

Feynman’s explanation of rubber elasticity, while intuitive appealing, is physically misleading in several aspects. His omission of entropy—or configurational disorder—is understandable because the concept is developed later in the chapter, but his mechanical imagery obscures the underlying physics. Describing polymer chains as “lengthwise” and attributing contraction to thermal agitations “on the sides of the chains” inaccurately portrays the molecular motion. Likewise, describing relaxation as molecules “striking” the chains and losing energy does not give the correct explanation for the associated temperature change. He could have distinguished two related but distinct thermoelastic phenomena: the contraction of a stretched rubber band when heated, and the temperature changes accompanying sufficiently rapid stretching or relaxation. The first is the Gough-Joule effect (Gough, 1805; Joule, 1859) ; the second is the associated adiabatic temperature response (or Adiabatic Gough-Joule effect). Both are clearly observed when the rubber is loaded, since an unloaded rubber shows little appreciable contraction upon heating.

 

An Alternative Explanation: Entropic Elasticity

To understand the effect better, we can model rubber as a cross-linked network of flexible polymer chains. In its relaxed state, thermal fluctuations allow the chains to have a vast number of random-coil conformations, corresponding to high configurational entropy (S). Stretching the rubber restricts the chains to more aligned, ordered conformations, reducing the number of accessible microstates and thereby decreasing the entropy. The tension can be understood in terms of the Helmholtz free energy F = U – TS, where U is the internal energy, and T is the absolute temperature. In short, the essential physics is not molecular “bombardment” but the temperature-dependent configurational entropy and Helmholtz free energy of the polymer chains. However, this entropic picture is an idealization: real elastomers depart from pure entropic elasticity at large strains due to bond distortions and possibly strain-induced crystallization, leading to hysteresis and other nonideal behavior.

 

2. Historical development

“At the time when Carnot lived, the first law of thermodynamics, the conservation of energy, was not known. Carnot’s arguments were so carefully drawn, however, that they are valid even though the first law was not known in his time!... So it has often been said that Carnot’s logic was wrong. But his logic was quite correct. Only Clapeyron’s simplified version, that everybody read, was incorrect (Feynman et al, 1963).”

 

Feynman's historical account of thermodynamics—particularly his discussion of Sadi Carnot and Émile Clapeyron—compresses several important historical distinctions in the interest of developing physical intuition. While this approach is pedagogically effective, it can leave the impression that Carnot’s theorem emerged fully formed or that Clapeyron was the first to assume that heat was conserved under the caloric theory. The actual history is more nuanced: Carnot’s central insight survived, but both its physical interpretation and mathematical formulation were substantially refined by later scientists. Engaging with this history—its false starts, conceptual conflicts, and brilliant insights—is one of the most effective ways to grasp the underlying physics.

 

Based on Feynman’s explanations, some may conclude that Carnot’s (1824) work is a curious mix: a correct principle derived from a fundamentally wrong premise. As Carnot (1824) wrote: “The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of the caloric (p. 20).” Strictly speaking, the modern Carnot theorem is a refined, quantitative, and logically reconstructed version of Carnot’s original result rather than a direct restatement of it. The following three points are worth clarifying.

 

1. Carnot’s caloric premise was wrong.

Working within the caloric theory, Carnot treated heat as an indestructible fluid flowing from a hot body to a cold body without being consumed. He distinguished between sensible heat, which changes a body's temperature, and latent heat, which accompanies a change of state at constant temperature. However, a major weakness of the caloric theory was its unsatisfactory account of heat generated by friction, attributing it to caloric forced out from  between atoms. Thus, the caloric premise had to be discarded.

 

2. Carnot’s original theorem was quantitatively incomplete.

Carnot established a significant result: no engine operating between two fixed temperature reservoirs can be more efficient than a reversible engine operating between the same reservoirs, regardless of the working substance. However, he did not derive the modern efficiency formula:

h = 1 - T_C/T_H

This formula required Kelvin’s absolute temperature scale and Clausius’s entropy, both were developed after Carnot’s death. Carnot’s original contribution is thus better regarded as a qualitative inequality rather than a complete quantitative efficiency law.

 

3. Carnot’s logic lacked modern mathematical rigor

While meticulous for its time, Carnot's reasoning falls short of contemporary standards of mathematical and physical rigor: it lacked the first law, modern concept of entropy, absolute temperature scale, and a rigorous calculus-based treatments. Modern proofs rely on both the first and second laws, cyclic processes, and mathematically establish the inequality for reversible engines. Thus, it is fair to say that Carnot’s mathematical formulation was incomplete and non-rigorous by modern standards, even though his original idea—reversibility and the temperature-dependent limit—was insightful.

 

Conclusions

In short, Carnot’s original work was correct in its central universality claim, yet based on a false caloric premise, resulted quantitatively incomplete without the absolute-temperature efficiency formula, and formulated without modern mathematically rigor. The current Carnot theorem preserves Carnot’s central insight while rebuilding it from the first and second laws of thermodynamics, absolute temperature, and entropy.

 

“But it turned out that Clapeyron assumed, not the conservation of energy in general, but that heat was conserved according to the caloric theory, which was later shown to be false... Only Clapeyron’s simplified version, that everybody read, was incorrect (Feynman et al, 1963).”

 

Clapeyron’s contribution

Émile Clapeyron’s (1834) memoir did more than provide a simplified restatement of Carnot’s work. By reformulating Carnot’s largely verbal reasoning into an analytical mathematical framework and introducing the pressure-volume indicator diagram, Clapeyron made thermodynamic concepts more accessible and helped enable further development. While Clapeyron did assume the caloric theory, but he was not its originator: Lavoisier (1783) introduced caloric as “igneous fluid” to explain thermal expansion and contraction. Carnot, however, used caloric for a different purpose—treating it like water falling from a high elevation (hot reservoir) to a low elevation (cold reservoir) to perform work. Most importantly, Clapeyron’s adapted Carnot’s theory and refined it into a mathematical framework that proved essential for the later formulations of modern thermodynamics.

Abandoning Caloric

In his landmark 1824 book, Reflections on the Motive Power of Fire, Carnot treated heat as a conserved, indestructible fluid. Yet Carnot also abandoned caloric and wrote explicitly in his notes: “When a hypothesis no longer suffices to explain phenomena, it should be abandoned. This is the case with the hypothesis which regards caloric as matter, as a subtle fluid (p. 68).” This may explain why Feynman, in his comment on Clapeyron rather than Carnot, remarked: “… it turned out that Clapeyron assumed, not the conservation of energy in general, but that heat was conserved according to the caloric theory.” Nevertheless, the conservation of heat can be regarded as a special case of energy conservation—formulated at a time when physicists did not yet know the nature of heat. In a similar sense, while modern conservation of energy provides a better framework, yet Feynman reminded us that we still do not really understand what energy is. Rather, Feynman specified energy as an abstract mathematical quantity—a number that remains invariant under all physical transformations.

 

3. First Law

“We can write this as follows: The heat Q put into the system, plus the W done on the system, is the increase in the energy U of the system; the latter energy is sometimes called the internal energy: Change in U = Q+W (44.1). The change in U can be represented as adding a little heat ΔQ and adding a little work ΔW : ΔU=ΔQ+ΔW, (44.2) which is a differential form of the same law (Feynman et al., 1963).”

 

I. A Brief History of the First Law of Thermodynamics:

It may come as a surprise that Feynman introduces thermodynamics through the Gough-Joule effect and historical discussions that seem more closely related to the Second law than the First law. However, examining how the First Law was historically formulated reveals why different disciplines use different conventions for heat and work.

 

Clausius: Heat Engines and Thermal Energy

When Rudolf Clausius wrote the First Law in the differential form:

dQ = dU + AdW,

he was analyzing heat engines and the relationship between heat and work. In his notations, another form of the equation is:

where R is a gas constant, t is the Celsius temperature, a » 273, and v is the specific volume. In these applications, the thermal energy supplied (Q) was the primary input controlled and measured by the experimenter. Writing thermal energy on the left emphasizes the question that naturally arises in heat engine experiments: “What happens to the thermal energy added to the system?” Expressing dQ as the subject shows the experimental accounting:

Thermal energy added = Change in Internal Energy + Work Done by the system

The equation is correct and useful provided that other forms of energy transfer are negligible.

Gibbs: Internal Energy and Heterogenous Substances

Josiah W. Gibbs approached thermodynamics through the equilibrium of material systems—heterogenous substances. His emphasis was not heat transfer, but the internal energy stored within the substances. In Gibbs’s notation, a fundamental equations relation was written as:

de = t dh - p dV,

where e is the internal energy, h is entropy, V is volume, t is temperature, and p is pressure. Gibbs also stated the energy conservation law in forms such as:

e' - e'' = W - Q.

The contrast with Clausius is primarily one of emphasis: Clausius focused on heat engine, while Gibbs focused on heterogenous substances. Both contributed fundamentally to the foundations of thermodynamics. The mathematical form of the First Law continued to evolve, e.g., Planck (1945) changed the sign of Q and wrote it as:

U₂ – U₁ = W + Q.

He adopted the convention that is now standard in chemistry. Planck also noted that while the First Law involves the idealization of a perfect or isolated system, truly isolated systems do not exist in nature.

 

II. Different Conventions: Chemistry versus Physics

Differences in how the First Law is written are not merely historical; they arise from the chosen sign convention for work and from the pedagogical or practical context in which the equation is introduced.

 

Why Chemistry textbooks often write DU = Q + W

In chemistry, the system is typically a chemical mixture contained in a flask. The convention commonly adopted is that both heat transfer into the system and work done on the system are positive. The primary guiding question is: How much energy is transferred into or out of the chemical system?

• Q > 0: Thermal energy via heat transfer enters the system.
• W > 0: Work is done on the system.

The First Law is therefore written as:

DU = Q + W.

 

For work done against an external pressure, W = -PdV.  During an expansion, DV > 0, so W < 0, meaning the system performs work on its surroundings, transferring energy outward. This convention is convenient for tracking internal energy changes in chemical reactions.

Why Physics/Engineering textbooks often write DU = Q - W (or DQ = DU + DW)

In many physics and engineering textbooks, particularly in the analysis of heat engines, work performed by the system is defined as positive:

• Q > 0: Thermal energy via heat transfer enters the system.
• W > 0: the system does work on its surroundings.

The First Law is then written as

DU = Q - W.

 

For work done against an external pressure, W = +PdV.  During an expansion at constant pressure, DV > 0, so W > 0: the system does work on its surroundings, reducing its internal energy.

Neither convention is inherently more correct. The essential requirement is to define the system, specify the signs of heat and work, and use the convention consistently. Some physicists advocated adopting the “chemistry convention” to reduce confusions.

 

III. Clarifying Heat, Internal energy, and Work

Discussions of thermodynamics become confusing when terms like heat, heat energy, and internal energy are used loosely or interchangeably. These terms refer to different physical concepts and should be distinguished carefully.

 

Careful Use of terminologies

Heat is a process of energy transfer due to a temperature difference. It is not a substance stored within a body. Instead of saying “the heat in a cup of water,” it is more precise to refer to the water’s thermal energy or, where appropriate, its internal energy.

Internal energy is a state function representing the total microscopic energy of a system (including thermal, chemical, and nuclear energy). It is not necessarily identical to what is informally called thermal energy.

Thermal energy is a less precise term, generally referring to energy associated with microscopic molecular motion and interactions. It should not be used as an exact synonym for internal energy in every context.

Heat energy is common in everyday speech and introductory teaching, but it blurs the distinction between energy stored in a system and energy transferred across its boundary.

 

Some physicists use the expression “heat is not a noun, but a verb”—emphasizing that heat is a mode of energy transfer, not an entity residing within a body. While daily language uses “heat” as both a noun and a verb, maintaining the operational distinction between heat as energy-transfer, internal energy as a state is vital for clarity.

 

Distinguishing Work: W_on versus W_by

To improve clarity on the sign convention for work, we may introduce subscripts.

Let W_on denote work done on the system, the First Law is:

DU = DQ + DW_on,   DW_on = -pDV

During compression (DV < 0), DW_on = -pDV > 0: positive energy enters the system.

Let W_by denote work done by the system, the First Law is:

DU = DQ - DW_by,   DW_by = +pDV

During compression (DV < 0), DW_by = pDV < 0: This negative sign in -DW_by allows DU > 0 when Q = 0.

In both cases, for a reversible processes,

DU = DQ -pDV

This means that physical reality is invariant regardless of which convention is adopted.

However, debates over terminology and convention may persist.

 

Key Takeaways:

1. Gough–Joule Effect

• The Thermoelastic Response: Heating a loaded rubber band causes it to contract; cooling causes it to expand. This is the Gough-Joule effect. Conversely, rapid stretching warms the rubber, and rapid relaxation cools it (the adiabatic temperature response).
• Entropic elasticity is the central concept to rubber’s restoring force. Rubber tends to retract when stretched primarily because stretching restricts the possible configurations of its polymer chains, reducing their configurational entropy. The restoring force F » -T(¶S/¶L)_T captures the essential physics more accurately than Feynman’s explanation based on molecular bombardment.

 

2. Carnot's Theory and Its Historical Context

• A lasting insight emerged from an incorrect physical framework. Carnot’s analysis, developed within the caloric theory, anticipates the modern result that no engine between two fixed temperatures can exceed the efficiency of a reversible engine.
• Clapeyron’s contribution: He did not merely produce a "simpler, incorrect" version; he reformulated Carnot's ideas analytically, introduced the indicator (PV) diagram, and made them more accessible to Kelvin and Clausius.
• Historical lesson: Correct physics can emerge from incorrect premises. The conservation of heat is replaced by conservation of energy. The resolution was to discard  the caloric premise while refining the mathematical approach and Carnot’s theorem.

 

3. First Law of Thermodynamics

• Feynman’s formula DU = Q + W adopts the chemistry/IUPAC convention. Here W is work done on the system. Many physics/engineering textbooks instead write DU = Q - W, where is work done by the system.
• The First Law is a statement of energy conservation. It says that the change in internal energy equals the thermal energy (“heat”) added plus the work done on the system (or minus the work done by the system).
• The Necessity of explicit Sign conventions: Specify whether W means work done on or by the system, and whether Q is thermal energy added to or removed from the system. Internal energy (U) is a state property, whereas heat (Q) and work (W) describe modes of energy transfer across the system boundary.

 

The Moral of the Lesson:

Muscle Mechanics, Thermal therapy, and Long-Term Relaxation

1. The Historical Analogy: Rubber Bands versus Living Muscle

In the 1940s and early 1950s, Nobel laureate Albert Szent-Györgyi developed influential models of muscle contraction that drew analogies to coiled springs and rubber-like elasticity. However, while muscles share certain passive elastic properties with rubber, the analogy has its limitations. Living muscle, by contrast, is an active, metabolically powered force-generating system. Its contractile force arises from ATP-dependent interactions between actin and myosin, making it far more than a rubber band. The model of rubber band involving Gough-Joule effect is only a useful conceptual comparison, not a complete explanation of muscle contraction or relaxation.

 

2. Thermoelasticity and Thermal Therapy

The effectiveness of stretching before physical activity remains debated in sports science because its effects depend on the stretching method, duration, and subsequent exercise. Research has shown that static stretching can temporarily reduce maximal muscle strength and power under some conditions (Shrier, 2004; Winchester et al., 2008). From a thermodynamic perspective, the adiabatic Gough-Joule effect implies that rapidly stretching a rubber band can raise its temperature, whereas rapidly releasing a stretched rubber band can produce a cooling effect. Living muscles and tendons also exhibit thermoelastic behavior, but their temperature change during stretching cannot be attributed to the Gough-Joule effect alone. In essence, stretching and releasing biological tissues may produce different thermal responses depending on loading rate, tissue properties, and experimental conditions. The Goule-Joule effect is therefore a possible model for understanding one aspect of muscle-tendon thermoelasticity, but its quantitative contribution to temperature changes during stretching remains insufficiently established.

 

The physiological effects of hot and cold therapies have been researched extensively, although the mechanisms underlying their therapeutic benefits remain partially understood. Cold-water immersion induces vasoconstriction and reduces pain perception, while its effects on inflammation vary with the protocol and temperature. Warm-water immersion and sauna exposure can promote vasodilation, alleviates perceived stiffness, and aids relaxation. Interpreting the evidence is complicated by variation among studies in water temperature, treatment duration, timing relative to exercise, and outcome measures. Many investigations rely on subjective ratings of perceived soreness or sensation rather than measurable improvements in neuromuscular function. Responses may also vary with body composition, gender, and the presence and nature of an injury. Thus, optimal treatment protocols and clinically meaningful benefits—particularly for heat therapy and contrast water therapy—remain uncertain. In short, thermal therapies are best regarded as adjuncts that may improve comfort and facilitate short-term recovery rather than a guaranteed tool for performance-enhancement.

 

3. Long-Term Relief: The “Dead Hang”

Beyond thermal therapy, regular movements, targeted strengthening, and appropriate stretching are essential for improving postures and achieving long-term musculoskeletal health. One possible exercise is the dead hang, in which a person hangs from a pull-up bar while supporting some or all of their body weight. Depending on the individual’s mobility, grip, and strength, hanging may challenge grip strength and the muscles that control shoulder position. A dead hang, carefully executed, can decompress the spine, stretch tight shoulder muscles, engage core stabilizers, and correct posture. Over time, it may serve as an excellent foundational practice for preventing or reducing both back and shoulder pain.

 

Overcoming the Hurdle: “Hard Choices, Easy Life”

Beginning a new exercise routine can be a difficult decision, particularly for people with chronic conditions, joint stiffness, or previous injuries. Avoiding movement may provide short-term comfort, but prolonged physical inactivity can lead to muscle loss, reduced strength and mobility. Confronting the discomfort of beginning a routine is “hard choice” that yields long-term health and resilience. Importantly, exercise must be performed carefully, progressively, and mindfully, especially when managing existing injuries. People with shoulder instability, significant joint pain, or relevant spine conditions should seek professional guidance before attempting full-body-weight hanging.

The broader principle is simple: small, consistent effort today can support greater strength, mobility, and independence tomorrow. With patience and sensible progression, the dead hang may become one useful element of a broader exercise routine. And if you decide to give it a try, remember this reminder: Just hang it there.

 

Ten Tips for a Safe and Effective Dead Hang

Source: Dead Hang Benefits: Shoulders, Grip, Spine & Who Should Skip

1. Choose a secure, accessible bar

Ensure the pull-up bar is firmly mounted, stable, and able to support your full body weight. If possible, set the bar height so that you can reach it without jumping, or use a stable platform to begin and end the hang. This reduces joint impact and allows you to release the bar safely.

2.      Optimize grip and hand protection

Begin with a slightly wider-than-shoulder overhand grip, wrapping your thumbs securely around the bar. If your hands are prone to blisters, consider using gloves to improve grip stability.

3.      Protect your neck and shoulder with supported hanging

If you have a history of shoulder impingement or joint instability, consult a sport therapist before trying full hangs. Start with foot-assisted hangs: Allow your toes or feet to touch the floor lightly to reduce your “effective weight.” This is especially vital for beginners who are overweight.

4.      Increase duration gradually

Begin with short hangs of 5–10 seconds if you are new to the exercise. As your grip strength and joint tolerance improve, gradually increase to 20–30 seconds. Experienced athletes can progressively extend hang durations to 60 seconds or longer.

5.      Schedule rest days

If you are new to hanging, start with two or three sessions per week. Even if you are physically fit, incorporate at least 1–2 rest days per week (e.g., taking Sundays off) to prevent muscle fatigue and allow muscle recovery.

6.      Maintain Body Alignment

Keep your body relatively still; avoid swinging or kicking. Your legs may remain straight or slightly bend. The goal is controlled stillness, which strengthens your core muscles and allows gentle spinal decompression.

7.      Active & Passive Hangs

A passive hang involves gripping a bar, keeping the body still, and allowing the shoulder muscles to be relatively relaxed, while the body weight enables the stretch. An active hang looks similar from a distance but involves actively pulling the shoulder blades slightly down and back, lifting your body by about 2 cm. You may start with passive hangs, progressing to active hangs as scapular control improves, or alternating between them.

8.      Breathe rhythmically

Avoid holding your breath while hanging. Maintain slow, deep, and regular breathing to keep your nervous system calm. Counting breaths can help track duration, and rhythmical music through headphones may also assist in pacing.

9.      Recognize Warning Signs

Stop immediately if you experience sharp pain, numbness, tingling, or unusual weakness in your shoulders. Rest, gently stretch, and consult a medical professional if neurological or joint symptoms persist. Never continue the exercise by taking painkillers.

10.  Support recovery with adequate nutrition

Eat a balanced diet containing sufficient protein to support muscle repair, tendon health, and grip recovery. Adequate energy intake, hydration, quality sleep are equally important for long-term progress.

 

Recommended Beginner Routine

• Volume: 1–3 sets per day
• Duration: 5–10 seconds per set
• Rest Interval: 60–90 seconds between sets
• Frequency: 2–3 times per week
• Load: Begin with your feet gently touching the ground.

(These are guidelines, not rigid targets.)

 

Beyond the Dead Hang: Building a Balanced Routine

Longer hangs and added external weight are not necessarily better. Once you can hang comfortably, complement the dead hang with other exercises to develop stability and mobility:

1.      Planks: Improve endurance and a stronger core.

2.      Push-ups: Strengthen the chest, shoulders, and triceps.

3.      Swimming: Offer whole-body, low impact cardiovascular exercise. If convenient, swim after a dead hang session.

4.      Resistance-band pulls: Train the upper-body (horizontal) pulling musculature.

The guiding principle is simple: progress gradually, prioritize comfort and control, and use the dead hang as one component of a balanced exercise program—not as a cure-all for back or shoulder problems. Practice patience; resolving muscle imbalance typically takes 3 to 6 months of consistent, mindful habit. Hanging is healing.

The Longevity Connection: Grip Strength

In a 2015 Prospective Urban Rural Epidemiology (PURE) study published in The Lancet, researchers measured handgrip strength in 139,691 adults across 17 countries and followed them for a median of four years. The findings indicate that grip strength is inversely associated with all-cause mortality—that is, it is a marker of longevity. Building your grip through exercises like the dead hang offers benefits that extend far beyond simple joint and muscle relief.

 

Review Questions:

1. Gough–Joule Effect: How would you  explain why a stretched rubber band can warm when stretched rapidly and cool when released rapidly, and why can heating a rubber band under a constant load cause it to contract?

2. Historical Development: How did Carnot’s original theory of heat engines, despite its reliance on the incorrect caloric theory, contribute to modern thermodynamics, and how did the work of Clapeyron, Kelvin, and Clausius refine his ideas into the modern Carnot theorem?

3. First Law of Thermodynamics: How does the First Law express the conservation of energy, and why must we specify the sign convention for work—e.g., DU=Q+W versus DU=Q-W—when applying it to a thermodynamic process?

 

References:

Carnot, S., Clapeyron, E., & Clausius, R. (1960). Reflections on the Motive Power of Fire: And Others Papers on the Second Law of Thermodynamics by E. Clapeyron and R. Clausius. Dover Publications.

Clausius, R. (1851). “On the Moving Force of Heat, and the Laws regarding the Nature of Heat itself which are deducible therefrom.” Philosophical Magazine, 4(2), pp. 1–21 and 102–119.

Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

Gibbs, J. W. (1878). On the equilibrium of heterogeneous substances. American journal of science, 3(96), 441-458.

Gough, J. (1805). A description of a property of Caoutchouc, or Indian rubber. Memories of the Literacy and Philosophical Society of Manchester, 1, 288-295.

Joule, J. P. (1859). On some thermo-dynamic properties of solids. Philosophical Transactions of the Royal Society of London, 149, 91-131.

Leong, D. P., Teo, K. K., Rangarajan, S., Lopez-Jaramillo, P., Avezum, A., Orlandini, A., ... & Yusuf, S. (2015). Prognostic value of grip strength: findings from the Prospective Urban Rural Epidemiology (PURE) study. The lancet, 386(9990), 266-273.

Shrier, I. (2004). Does stretching improve performance?: a systematic and critical review of the literature. Clinical Journal of sport medicine, 14(5), 267-273.

Winchester, J. B., Nelson, A. G., Landin, D., Young, M. A., & Schexnayder, I. C. (2008). Static stretching impairs sprint performance in collegiate track and field athletes. The Journal of Strength & Conditioning Research, 22(1), 13-19.