Force and Energy

1. Force

1.1 Aristotle and the 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[2a], 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[2b].

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].

1.2 Archimedes and torque

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].

Diagram showing how torques balance on balancing scales or a seesaw.

A lever in balance. Image credit: Helen Klus/CC-NC-SA.

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

Torque = Force × Distance from pivot.

Here, Torque = 𝛕, Force = F, and Distance = d, so that:

𝛕 = Fd.

Here, the force is caused by the object's weight, which was later shown to be due to the force of gravity.

1.3 Archimedes and 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 greater than the force of gravity, then an 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.

Image credit: Helen Klus/CC-NC-SA.

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 to displace more fluid, which it can do by increasing in volume.

If an object increases in volume, then it will exert less pressure on the fluid because:

Pressure =
Force/Surface area
.

Here, Pressure = P, and Surface area = A so that:

P =
F/A
.

If an object increases in volume while remaining the same weight, then it will also become less dense because:

Density =
Mass/Volume
,

Here, Density = ρ, Mass = m, and Volume = V so that:

ρ =
m/V
.

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[8].

An aeolipile is a sphere placed above a container of water and positioned so that it can rotate on its axis. The container of water is heated and steam rises through tubes attached to the sphere. The steam is allowed to escape through holes on the top and bottom of the sphere, causing it to rotate as the steam escapes.

Animation of an aeolipile.

An aeolipile. Image credit: Michael Frey/CC-SA.

Hero used the power of steam to create automated machines, which were used to put on plays.

1.5 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.

Drawing of a parachute by Da Vinci.

Da Vinci’s Parachute. Image credit: Leonardo da Vinci/Public domain.

Da Vinci utilised the laws of friction[9], momentum[10], and centripetal force[11] - the inwards force that makes an object move in a circle. This is supplied, for example, by the string if you swing a yo-yo around.

Da Vinci also described capillary action[12]. 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.

The forces on molecules of liquid, which cause surface tension. Image credit: Booyabazooka/Public domain.

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 chemical bonding, opposite charges, or a variety of other reasons including mechanical bonding, which is used in Velcro.

1.6 Simon Stevin and ramps

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[13a].

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[13b].

Stevin imagined three planes - flat two-dimensional surfaces - that form a triangle. A loop of string is draped over them with beads attached, which are placed equally apart. It is balanced due to the tension in the string at the top, point T on the picture below.

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

Forces on a ramp by Stevin. Image credit: Simon Stevin/Public domain.

If you removed the beads below the bottom of the triangle, at points S and V, then it would still balance despite the fact that there are more beads on the longer slope. This shows that the amount of force you need to push something up a slope is reduced the longer, and hence the less steep, the slope.

1.7 Newton's laws and the Coriolis effect

French mathematician Gustave-Gaspard Coriolis was the first to explain the science behind the Coriolis effect. This shows that Coriolis and centrifugal forces are evident if you transform English natural philosopher Isaac Newton's laws of motion on to a rotating reference frame.

The centrifugal force is the outwards force caused by rotation. The Coriolis force refers to the fact that the path of objects appears deflected as they travel over a rotating body. This is because the reference frame moves into the path of the object. The amount that the frame moves depends where you are on that frame.

In 1835, Coriolis published this theory in relation to machines with rotating parts, like water wheels, but it also applies to the path of an object travelling across the surface of the Earth[14].

Animation of the Coriolis force

Illustration of the Coriolis force. Image credit: modified by Helen Klus, original image by Hubi/CC-SA.

Diagram of the Earth showing the wind moving clockwise in the northern hemisphere, and anticlockwise in the southern hemisphere.

Diagram showing how wind would be deflected if only the Coriolis force is present. Image credit: Kes47/CC-SA.

An object would land slightly westwards of its target if it were fired from the North Pole to the equator. This is because the Earth is moving from west to east. It would land slightly eastwards of its target if it were fired northwards, from the equator towards the North Pole.

Large-scale objects, like air in the atmosphere and water in the ocean, deflect to the east as they travel towards the pole and west as they travel towards the equator. This results in clockwise circular motion in the northern hemisphere, and counter clockwise motion in the Southern hemisphere.

2. Force and acceleration

2.1 Early forms of Newton’s laws

Iranian polymath Abū Rayḥān al-Bīrūnī realised that acceleration is related to non-uniform motion in about 1021[15]. Acceleration is the rate of change of an object's velocity, where velocity is a measure of an object's speed in a given direction.

Al-Bīrūnī also noted that everything on Earth seems to be attracted to the Earth's centre[16], and was one of the first people to suggest that friction, the force that resists motion, can cause heat[17].

About 100 years later, Iraqi philosopher Abu'l-Barakāt al-Baghdādī discovered that force is proportional to acceleration[18]. Both were precursors to Newton's 2nd law of motion, which was published in 1687.

Newton's 2nd law of motion states that:

Force = Mass × Acceleration.

Here, Acceleration = a so that:

F = ma.

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[19]. This was a precursor to Newton's 3rd law of motion - which states that all forces act in pairs that are equal in magnitude and opposite in direction.

2.2 Galileo's 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.

Galileo's student, Italian mathematician Vincenzo Viviani, stated that in 1589 Galileo had dropped balls of different weights from the Leaning Tower of Pisa[20]. He did 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 is 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 is 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 distance the balls travelled was proportional to the square of the time it took them to fall, something that had first been discovered by French mathematician Nicole Oresme in the 14th century[22].

The fact that different masses fall at the same rate was later explained by combining Newton’s 2nd law of motion with his theory of universal gravitation, and proven when Apollo 15 astronaut Commander David Scott dropped a feather and hammer at the same time in 1971. The Moon has no atmosphere to create air resistance, and so they fell at the same rate, reaching the Moon's surface at the same time.

Gif showing a feather and hammer being dropped on the Moon and landing at the same time.

Commander David Scott dropping a feather and hammer on the Moon in 1971. Image credit: NASA/Public domain.

In 1602, Galileo became the first person to suggest using a pendulum - the first harmonic oscillator - to measure time. The laws of harmonic oscillators were later devised by Dutch natural philosopher Christiaan Huygens and English natural philosopher Robert Hooke.

2.3 Galileo's relativity

In 1632, Galileo 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[23a].

This influenced Newton's 1st 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.

Galileo also suggested that we may not feel the motion of the Earth. This is because all motion is relative and there is no absolute state of rest. When you are on a ship, for example, you are at rest with respect to the floor even though you are moving with respect to the sea. If the sea is calm, it may be impossible to tell if you are moving or stationary without an outside reference, like a window to look out of.

Galileo stated:

"Shut yourself up with some friend in the main cabin below decks on some large ship, and have with there flies, butterflies, and other small flying animals. Have a large bowl of water with some fish in it; hang up a bottle that empties drop by drop into a wide vessel beneath it. With the ship standing still, observe carefully how the little animals fly with equal speed to all sides of the cabin. The fish swim indifferently in all directions; the drops fall into the vessel beneath; and, in throwing something to your friend, you need throw it no more strongly in one direction than another, the distances being equal; jumping with your feet together, you pass equal spaces in every direction. When you have observed these things carefully (though there is no doubt that when the ship is standing still everything must happen in this way), have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still"[23b].

Galileo showed that relative velocities are additive. This means that if you walk at velocity v across the deck of a ship that is moving at velocity u, then your velocity with respect to someone standing on the shore s, is equal to v + u.

This shows that it's faster to walk on a moving walkway because your total speed will equal the speed you are walking plus the speed of the walkway. These observations influenced German-Swiss-American physicist Albert Einstein's theory of special relativity, published in 1905.

3. Force and momentum

Alexandrian philosopher John Philoponus had 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:

Momentum = Mass × Velocity.

Here, Momentum = p, and Velocity = v so that:

p = mv.

French natural philosopher Rene Descartes devised a similar equation in the 1600s[26].

Momentum is related to an objects inertia, which is related to its mass and defined as the tendency of an object to resist a change in motion.

English mathematician John Wallis was one of the first people to suggest that momentum is conserved during collisions in 1668, along with 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.

Diagram showing the momentum of two separate objects. Momentum equals mass × velocity.
Diagram showing the total momentum after two objects collide.

The conservation of momentum. Image credit: Helen Klus/CC-NC-SA.

Newton also considered angular momentum, which describes objects that are moving in a circle[28]:

Angular momentum = Mass × Velocity × Radius.

Here, Angular momentum = L, and Radius = r so that:

L = mvr.

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.

4. Force and energy

In 1638, Galileo became the first person to suggest that simple machines might not create energy; they may only transform it from one form to another[29a].

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[29b].

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].

In 1829, Coriolis showed that:

Kinetic energy = ½ × Mass × Velocity2.

Here, Kinetic energy = KE so that:

KE = ½mv2[31].

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[29c].

The term 'energy' was first used to describe kinetic energy by British natural philosopher Thomas Young in 1807, and was popularised by British natural philosopher William Thomson, better known as Lord Kelvin, and British engineer William Rankine, in the 1850s[32a][33].

Rankine coining the term 'potential energy' in 1853[32b], where:

Potential energy = Mass × Acceleration due to gravity × Height.

Here, Potential energy = PE, Acceleration due to gravity = g, and Height = h so that:

PE = mgh.

5. Modern theories

In the 20th century, forces were explained in terms of Einstein's general relativity and quantum field theories, and energy was explained in terms of special relativity and quantum mechanics.

6. References

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

  2. (a, b) Aristotle and Stocks, J. L. (trans), 2015 (350 BCE), 'On the Heavens', eBooks@Adelaide.

  3. Plato and Jowett, B. (trans), 2009 (360 BCE), 'Timaeus', The Internet Classics Archive.

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

  5. Archimedes and Heath, T. L. (ed), 1897 (260 BCE), 'On the Equilibrium of Planes' in 'The Works of Archimedes', Cambridge University Press.

  6. Paipetis, S. A. and Ceccarelli, M., 2010, 'The Genius of Archimedes: 23 Centuries of Influence on Mathematics, Science and Engineering', Springer Science & Business Media.

  7. Archimedes and Heath, T. L. (ed), 1897 (250 BCE), 'On Floating Bodies' in 'The Works of Archimedes', Cambridge University Press.

  8. Papadopoulos, E. and Ceccarelli, M. (ed), 2007, 'Heron of Alexandria (c. 10–85 AD)' in 'Distinguished Figures in Mechanism and Machine Science', Springer.

  9. Pitenis, A. A. and Dowson, D. and Sawyer, W. G., 2014, 'Leonardo da Vinci’s friction experiments: An old story acknowledged and repeated', Tribology Letters, 56, pp.509-515.

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

  11. Campbell, S., 'The Ingenuity behind Leonardo Da Vinci Inventions', Leonardo Da Vinci's Life, last accessed 15-02-16.

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

  13. (a, b) Roux, S. and Festa, E., 2008, 'Mechanics and Natural Philosophy before the Scientific Revolution', Kluwer Academic Publishers.

  14. Coriolis, G. G., 1835, 'Sur les Équations du Mouvement Relatif des Systèmes de Corps', Journal de l’Ecole Royale Polytechnique, 15, pp.144-154.

  15. Afridi, M. A., 2013, 'Contribution of Muslim Scientists to the World: An Overview of Some Selected Fields', Revelation and Science, 3, pp.40-49.

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

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

  18. Gari, L. and Abattouy, M., 'Abu ‘l-Barakat al-Baghdadi: Outline of a Non-Aristotelian Natural Philosophy', Muslim Heritage, last accessed 15-02-16.

  19. Franco, A. B., 2004, 'Avempace, projectile motion, and impetus theory', Journal of the History of Ideas, 64, pp.521-546.

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

  21. Berghe, G. V., Devreese, J. T. and Hockey, T. et al. (eds), 2014, 'Stevin, Simon' in 'Biographical Encyclopedia of Astronomers', Springer New York.

  22. Nicodemi, O., 2010, 'Galileo and Oresme: Who Is Modern? Who Is Medieval?', Mathematics Magazine, 83, pp.24-32.

  23. (a, b) Galilei, G. and Drake, S. (trans), 1953 (1632), 'Dialogue Concerning the Two Chief World Systems, Ptolemaic and Copernican', Modern Library.

  24. Sayili, A., King, D. A. (ed) and Saliba, G. (ed), 1987, 'Ibn Sina and Buridan on the Motion of the Projectile' 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', New York Academy of Sciences.

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

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

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

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

  29. (a, b, c) New World Encyclopedia, 'Conservation of energy', last accessed 15-02-16.

  30. Arthur, R. T. W., 2014, 'Leibniz', John Wiley & Sons.

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

  32. (a, b) Vieil, E., 2012, 'Understanding Physics and Physical Chemistry Using Formal Graphs', CRC Press.

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

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