Chapter 4. Force, Momentum, and Energy

4.1 Force

4.1.1 Aristotle and the five elements

The concept of force has been used since the first civilisations invented simple machines, like levers and ramps. Simple machines allow less force to be used to do the same amount of work.

Ancient Greek philosopher Empedocles first suggested that there are four elements - water, earth, fire, and air - in about 475 BCE. Ancient Greek philosopher Aristotle popularised this idea in about 350 BCE.[1] Aristotle also claimed that space contains a fifth element, the aether,[2] an idea first suggested by Ancient Greek philosopher Plato about 10 years earlier.[3]

Aristotle claimed that all of these elements have a natural state, and that they will tend towards this state if left alone. Aristotle believed that it’s natural for water and earth to be motionless on the ground, for example, and that unnatural motion is required to move them. Aristotle argued that this requires a force to be applied, and that it must continue to be applied as long as the object is not in its natural state.[2] Aristotle suggested that in the case of objects like arrows, which continue to stay above the ground even when nothing is touching them, displaced air must provide the force.[4]

4.1.2 Archimedes and the law of the lever

Ancient Greek philosopher Archimedes of Syracuse studied how pulleys, screws, and levers work in about 260 BCE.[5] Levers include objects like wheelbarrows and tongs. Archimedes showed that a small weight could balance a larger one if the smaller weight were placed further from the pivot. This is why he is quoted as saying:

“...give me a place to stand on, and I will move the earth”.[6]

We would now say that a lever will balance if the torques on both sides of the pivot are equal, where:

τ = Fd = mgd (4.1)

Here, τ is the torque, F is the force, and d is the distance from the pivot. The force is due to the object’s weight (mg), where m is mass and g is acceleration due to gravity. Weight was later shown to be a description of the force of gravity (discussed in Chapter 5).

g will be the same for both sides, and so for a lever in balance:

m1d1 = m2d2 (4.2)
Diagram showing how torques balance on balancing scales or a seesaw.

Figure 4.1
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A lever remains in balance if the torques on both sides are the same, where: torque = mass × distance from the pivot.

4.1.3 Archimedes’ principle of buoyancy

Archimedes also discovered that if an object is immersed in a fluid, like water or air, then it will experience an upwards force, known as buoyancy.[7] For an object to float, the buoyant force must be equal to, or greater than, the weight of the object. If the weight of the object is greater than the buoyant force, then the object will sink.

Diagram showing that if the force of weight is less than, or equal to, the force of gravity, then an object will float.

Figure 4.2
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If the weight of an object is less than the force due to buoyancy, the object will float.

Diagram showing that if the force of weight is greater than the force of gravity, then an object will sink.

Figure 4.3
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If the weight of an object is greater than the force due to buoyancy, the object will sink.

The force of buoyancy is equal to the weight of the fluid that the object displaces. This means that to increase buoyancy, an object needs to either lose weight or displace more fluid, which it can do by increasing in volume (V). If an object increases in volume, then it will also increase in surface area (A). This means it will exert less pressure (Pr) on the fluid because:

Pr = F/A (4.3)

If an object increases in volume while remaining the same mass (m), then it will also decrease in density (ρ) because:

ρ = m/V (4.4)

4.1.4 Hero and the first steam engine

In about 60 CE, Ancient Greek engineer Hero of Alexandria described six simple machines: the lever, the windlass (a type of winch), the pulley, the wedge, the screw, and a primitive steam engine called an aeolipile.

An aeolipile is composed of a sphere that is placed above a container of water and positioned so that it can rotate on its axis. When the container of water is heated, steam rises through tubes attached to the sphere. The steam is allowed to escape through holes on the top and bottom of the sphere, and this causes it to rotate. Hero used the power of steam to create automated machines, which he used to put on plays.[8]

Diagram of an aeolipile.

Figure 4.4
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An aeolipile is a simple steam engine, which was first created by Hero of Alexandria in about 60 CE.

4.1.5 Early forms of Newton’s laws

Iranian polymath Abū Rayḥān al-Bīrūnī realised that acceleration is related to non-uniform motion in about 1021.[9] Al-Bīrūnī also noted that everything on Earth seems to be attracted to the Earth’s centre,[10] and was one of the first people to suggest that friction, the force that resists motion, can cause heat.[11] About 100 years later, Iraqi philosopher Abu’l-Barakāt al-Baghdādī discovered that force is proportional to acceleration,[12] a precursor to Newton’s second law of motion (discussed in Chapter 5).

Andalusian polymath Ibn Bâjjah (also known as Avempace) was the first to suggest that for every force there is a reaction force in around 1120.[13] This was a precursor to Newton’s third law of motion (also discussed in Chapter 5).

4.1.6 Leonardo da Vinci and capillary action

Italian polymath Leonardo da Vinci designed a number of advanced machines, including a helicopter, an aeroplane, a tank, a parachute, and a hang glider, in the late 1400s, although not all of them were built. Da Vinci’s machines used levers, pulleys, gears, and cranks - an arm attached at right angles to a rotating shaft, like those used to manually open a car window.

Da Vinci utilised the laws of friction,[14] momentum,[15] centripetal force[16] - the force that causes rotating objects to move in a circle, and capillary action.[17] This is the ability of water to flow against gravity in narrow spaces, like when liquid is drawn up between the hairs of a paintbrush. Capillary action occurs because of surface tension and adhesive forces.

Surface tension can be seen when something denser than water floats on the surface. This extra surface pressure occurs because the atoms on the surface are pulled inwards. This is because, unlike all the other atoms in the liquid, they do not have atoms above them to balance the force of those below.

Diagram showing molecules on the surface of water experience an uneven force.

Figure 4.5
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The forces on molecules of liquid, which cause surface tension.

Adhesion is the tendency of dissimilar surfaces to cling to each other, like dew attached to a spider’s web. Adhesion can be caused by many things, including chemical bonding, opposite charges, and mechanical bonding.

4.1.7 Simon Stevin’s inclined plane

Italian mathematician Jordanus de Nemore had already shown that the ramp could be described as a simple machine in the 13th century. He did this by describing why moving an object up a slope requires less force than lifting it straight up.[18]

This idea did not become popular until the 1580s, when people such as Flemish mathematician Simon Stevin and Italian natural philosopher Galileo Galilei published their own proofs.[18]

Stevin imagined a loop of string with weights attached that are placed equally apart. If the string is placed over a small ramp, it will balance due to the tension in the string at the top, point T in Figure 4.6.

Diagram showing that moving an object vertically requires less force if you use a ramp.

Figure 4.6
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Forces on a ramp by Stevin.

If you cut the string below the bottom of the triangle, at points S and V in Figure 4.6, then the string will still balance despite the fact that there is more weight on the longer slope. There must be an equal force on both sides for the string to balance, and so it takes less force per mass to push something up a longer, and hence less steep, slope.

4.1.8 Galileo and the laws of motion

Galileo was one of the first modern scientists to state that the laws of nature are mathematical. He made observations and then tried to determine the mathematics that explained them. In 1623, he stated that:

“Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth”.[19]

Galileo’s student, Italian mathematician Vincenzo Viviani, claimed that Galileo had dropped balls of different weights from the Leaning Tower of Pisa in 1589.[20] He is said to have done this in order to demonstrate that they would fall at the same rate as long as air resistance was negligible. This contradicted Aristotle’s belief that more massive objects would fall faster. Although there’s no proof that Galileo performed this experiment, Stevin may have performed the experiment from the church tower in Delft in the Netherlands in 1586.[21]

There’s evidence that Galileo did conduct experiments to prove that bodies fall at the same rate whatever their mass, but he did this by timing how long it took balls to roll down a ramp. Galileo showed that the time it took the balls to fall was proportional to the square root of the distance they travelled.

Δt ∝ √d  (4.5)

This was something that had first been discovered by French mathematician Nicole Oresme in the 14th century,[22] and was analogous to the equation linking the period and length of pendulums (discussed in Chapter 2). Galileo became the first person to suggest using a pendulum to measure time in 1602.

The fact that different masses fall at the same rate was later explained by combining Newton’s second law of motion with his theory of universal gravitation. This was proven in 1971, when Apollo 15 astronaut Commander David Scott dropped a feather and hammer at the same time on the Moon (discussed in Chapter 5).

In 1632, the same year Galileo published his theory of relativity (discussed in Chapter 3), he contradicted Aristotle once again by arguing that once in motion, objects will remain in motion, travelling in the same direction at a constant speed unless they are acted on by an outside force, like friction.[23] This influenced Newton’s first law of motion, which states that objects continue to move in a state of constant velocity, which can be zero, unless acted upon by an external force.

Newton published his laws of motion and universal gravitation in 1687 (discussed in Chapter 5). In the 20th century, forces were explained in terms of Einstein’s theory of general relativity (discussed in Chapter 8) and quantum field theories (discussed in Book II).

4.2 Momentum

Alexandrian philosopher John Philoponus criticised Aristotle’s theory of motion in the 6th century. Philoponus modified Aristotelian physics to account for the motion of arrows by stating that a hurled object acquires a temporary motive power.[24]

In the mid-1300s, French philosopher Jean Buridan improved upon Philoponus’ ideas with his theory of impetus. Buridan described impetus as a force that enables an object like an arrow to continue moving in the direction it is fired. This force is opposed by air resistance and the pull of the Earth, which is why the arrow does not fly forever.

Buridan stated that impetus is equal to weight multiplied by velocity[25] - an early form of the equation for momentum (p):

p = mv (4.6)

French natural philosopher Rene Descartes devised a similar equation in the 1600s.[26] English mathematician John Wallis was one of the first people to suggest that momentum is conserved during collisions in 1668, along with Dutch natural philosopher Christiaan Huygens, and English architect and astronomer Christopher Wren.[27] This means that the total momentum of two objects will be the same before and after a collision.

Newton proved this, and showed the link between force and momentum in 1687 (discussed in Chapter 5). Newton also considered angular momentum (L), which describes objects that are moving in a circle:[28]

L = pr (4.7)

The conservation of angular momentum means that the angular momentum of a spinning object will remain the same if the radius decreases. This means that either the mass or velocity must increase, and explains why ice skaters spin faster when they hold their arms close to their body, as well as Kepler’s second law (discussed in Chapter 3), which states that planets orbit at a higher velocity the closer they are to the Sun.

The conservation of momentum

The conservation of momentum shows that the total momentum of two objects will be the same before and after a collision.

If two objects are moving in the same direction:

Diagram showing the momentum of two objects before and after a collision. Momentum equals mass × velocity.

Figure 4.7 Image credit

p = 20 × 10 p = 5 × 30
p = 200 kg km/s p = 150 kg km/s p = 200 + 150
p = 350 kg km/s
p = mv
and so
v = p/m v = 350/25
v = 14 km/s

If two objects are moving in opposite directions:

Diagram showing the total momentum after two objects collide while moving towards each other.

Figure 4.8 Image credit

p = 20 × 10 p = 5 × -30
p = 200 kg km/s p = -150 kg km/s p = 200 - 150
p = 50 kg km/s
v = 50/25
v = 2 km/s

4.3 Energy

Forces transfer energy in a process known as ‘work’, where:

ΔE = W = Fd (4.8)

Here, E is energy and W is work.

Galileo became the first person to suggest that simple machines might not create energy - they only transform it from one form to another, in 1638.[29] German mathematician Gottfried Leibniz devised the first mathematical theory of the conservation of energy in the 1670-1680s. Leibniz showed that a force known as vis viva, which is Latin for living force, is conserved during collisions.[29]

In 1740, French mathematician Émilie du Châtelet combined the theories of Leibniz and Dutch natural philosopher Willem ’s Gravesande, to show that the energy of a moving object is proportional to the square of its velocity.[30] Finally, in 1829, French mathematician Gustave-Gaspard Coriolis showed that:[31]

KE = 1/2mv2 (4.9)

Here, KE is the kinetic energy, the energy of a moving object. The equation for kinetic energy can be derived from English natural philosopher Isaac Newton’s second law of motion, which was published in 1687 (discussed in Chapter 5).

German natural philosophers Julius Robert von Mayer and Hermann von Helmholtz, and British natural philosopher James Prescott Joule formed the law of conservation of energy in the 1840s, although their theory was stated in terms of vis viva rather than energy.[29]

The term ‘energy’ was first used by British natural philosopher Thomas Young in 1807, in reference to kinetic energy. The term was popularised by British natural philosopher William Thomson, better known as Lord Kelvin, and British engineer William Rankine, in the 1850s.[32,33] Rankine coined the term ‘potential energy’ in 1853.[32] Gravitational potential energy (GPE) refers to the energy that’s needed to keep an object from falling from a height (h), where:

GPE = mgh (4.10)

Here, (g) is acceleration due to gravity (first discussed in Chapter 2).

An object like a pendulum converts gravitational potential energy into kinetic energy as it swings back and forth. Potential energy and the law of pendulums were later explained by Newton’s theory of gravitation, which was published in 1687 (discussed in Chapter 5).

In the 20th century, energy was explained in terms of Einstein’s theory of special relativity (discussed in Chapter 7) and quantum mechanics (discussed in Book II).

4.4 References

  1. Aristotle, On the Generation and Corruption, translated by Joachim, H. H., eBooks@Adelaide, 2015 (350 BCE).

  2. Aristotle, On the Heavens, translated by Stocks, J. L., eBooks@Adelaide, 2015 (350 BCE).

  3. Plato, Timaeus, translated by Jowett, B., The Internet Classics Archive, 2009 (360 BCE).

  4. Machuga, R., Life, the Universe and Everything: An Aristotelian Philosophy for a Scientific Age, Casemate Publishers, 2012.

  5. Archimedes in The Works of Archimedes, (Ed.: Heath, T. L.), Cambridge University Press, 1897 (260 BCE).

  6. Hippocrates, Galen, The Writings of Hippocrates and Galen, translated by Redman Coxe, J., Lindsay and Blakiston, 1846.

  7. Archimedes in The Works of Archimedes, (Ed.: Heath, T. L.), Cambridge University Press, 1897 (250 BCE).

  8. Papadopoulos, E. in Distinguished Figures in Mechanism and Machine Science, (Ed.: Ceccarelli, M.), Springer, 2007.

  9. Afridi, M. A., Revelation and Science 2013, 3, 40–49.

  10. Bagheri, M., The Influence of Indian Mathematics and Astronomy in Iran, National Institute of Advanced Studies, 2001.

  11. Nasr, S. H., An Introduction to Islamic Cosmological Doctrines, SUNY Press, 1993.

  12. Gari, L., Abattouy, M., Abu ‘l-Barakat al-Baghdadi: Outline of a Non-Aristotelian Natural Philosophy, Muslim Heritage.

  13. Franco, A. B., Journal of the History of Ideas 2004, 64, 521–546.

  14. Pitenis, A. A., Dowson, D., Sawyer, W. G., Tribology Letters 2014, 56, 509–515.

  15. Keele, K. D., Leonardo Da Vinci’s Elements of the Science of Man, Academic Press, 2014.

  16. Campbell, S., The Ingenuity behind Leonardo Da Vinci Inventions, Leonardo Da Vinci’s Life.

  17. Russell, J. S., Perspectives in Civil Engineering: Commemorating the 150th Anniversary of the American Society of Civil Engineers, ASCE Publications, 2003.

  18. Roux, S., Festa, E., Mechanics and Natural Philosophy before the Scientific Revolution, Kluwer Academic Publishers, 2008.

  19. Galilei, G. in Discoveries and Opinions of Galileo, (Ed.: Drake, S.), translated by Drake, S., New York: Doubleday & Co, 1957 (1623).

  20. Moody, E. A., Studies in Medieval Philosophy, Science, and Logic, University of California Press, 1975.

  21. Berghe, G. V., Devreese, J. T. in Biographical Encyclopedia of Astronomers, (Eds.: Hockey, T., Trimble, V., Williams, T. R., Bracher, K., Jarrell, R. A., Marché, J. D. I., Palmeri, J., Green, D. W. E.), Springer New York, 2014.

  22. Nicodemi, O., Mathematics Magazine 2010, 83, 24–32.

  23. Galilei, G., Dialogue Concerning the Two Chief World Systems, Ptolemaic and Copernican, translated by Drake, S., Modern Library, 1953 (1632).

  24. Sayili, A. in From Deferent to Equant: A Volume of Studies in the History of Science in the Ancient and Medieval Near East in Honor of E.S. Kennedy, (Eds.: King, D. A., Saliba, G.), New York Academy of Sciences, 1987.

  25. Pedersen, O., Early Physics and Astronomy: A Historical Introduction, CUP Archive, 1993.

  26. Cottingham, J., The Cambridge Companion to Descartes, Cambridge University Press, 1992.

  27. Anstey, P., Jalobeanu, D., Vanishing Matter and the Laws of Nature: Descartes and Beyond, Routledge, 2010.

  28. Smith, J. O., Physical Audio Signal Processing, W3K Publishing, 2010.

  29. New World Encyclopedia, Conservation of energy, New World Encyclopedia, 2013.

  30. Arthur, R. T. W., Leibniz, John Wiley & Sons, 2014.

  31. Krehl, P. O. K., History of Shock Waves, Explosions and Impact: A Chronological and Biographical Reference, Springer Science & Business Media, 2008.

  32. Vieil, E., Understanding Physics and Physical Chemistry Using Formal Graphs, CRC Press, 2012.

  33. Kokkotas, P. V., Malamitsa, K. S., Rizaki, A. A., Adapting Historical Knowledge Production to the Classroom, Springer Science & Business Media, 2011.

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