S1C10: Gravitational & Mechanical Energies

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a) Making sense of gravity

For those who have read Chapter 9, we concluded with the topic of gravitational radiation, so the beginning of this last chapter should provide some continuity as we tackle the last form of energy: gravitational potential energy; I use the word “last” since mechanical energy, which we will also discuss later, should be thought of as a meta-form of energy, consisting of the aggregate of potential and kinetic energy in a system, rather than being tied to a definite fundamental interaction or behaviour of matter.

However, before we move on to the potential energy aspect, I think it would be a missed opportunity and perhaps even a misstep not to re-engage with the concept of gravity itself. If anything, it will give all readers some food for thought and exhibit a few of the inherent difficulties in conceptualizing certain properties of the physical universe, tying macroscopic-level phenomena with the sub-atomic makeup and laws.

In classical physics, that is at your bread-and-butter non-relativistic speeds, non-high energy levels and supra-atomic scale, gravity is a fundamental interaction responsible for the attraction between objects with mass. Read carefully: interaction. This means if there are two objects and one is more massive than the other, each one will pull on the other. If there are more than two objects, each one of them will display some pull on every single of the others as well. So when we stand on the Earth’s ground, not only are we attracted towards the centre of mass of the planet somewhere in its metallic inner core, but we also exert a gravitational pull on the planet. Considering our respective masses, this does not make much of a difference in terms of the shape of the planet; different story when it is the Moon or the Sun.

Indeed, the tides of the oceans are primarily caused by the moon’s gravitational interaction with the water molecules, and not by the Sun because it is nearly 400 times more distant so its gravitational field gradient is much weaker. Gradient is the rate of change of the field strength, it changes proportionally to the cube of the distance, not the square like the strength of the gravitational field. You may have heard about the tidal bulge or seen a graphical representation of two bulges, one on the moon side and the other on the opposite side. Well, I could not comprehend this explanation (which even the NASA website uses!) and of course, this doesn’t accord with the facts, the regional to and fro exhibited by tides. And so I dug deeper until I landed on a coherent online explanation titled “there is no tidal bulge”; the link is provided at the bottom of the chapter. The Earth is not an idealized system and many factors have an influence on tides, the underwater topography in particular, the land mass rupturing the flow from one ocean to the next, and the deflection created by the Earth rotation known as the Coriolis effect. As a consequence of all these variables and constraints, the main dynamics of tides actually occur around nodes with zero tidal amplitude called amphidromic points. If you search for the term amphidromic on the internet, you will see a coloured map where the points appear in dark blue – this map is also shown in the article I provided the link to.

As for the Earth’s pull on the Moon it can obviously be witnessed in the fact that the latter is in orbit around the former. The Earth being so much more massive, it is the dominant object in the Earth-Moon system and is considered the planet and the Moon the satellite. And no, technically it is not a binary planet system because the system’s centre of mass, the barycentre, would have to be outside of both bodies for it to be the case according to one of the often-used criteria, and it happens to be well within the Earth’s volume. For a double planet system to form there would need to be much less difference between the masses of the two bodies and in the case of Moons Vs Earth the ratio is about 0.122. Besides capturing Moon into its orbit, Earth’s gravitational pull also results in what is known as tidal locking, the synchronous rotation of the Moon on itself and its revolution around the Earth. This explains why we always see the same side of the Moon and the far side remains unobservable without resorting to some kind of space travel.

Similarly, the Sun’s gravitational pull on the Earth being so much stronger than the opposite attraction, it is the latter which orbits the former – at 274 m/s², the surface gravity of the Sun is about 28 times stronger than the average one on Earth of 9.8 m/s², an acceleration level we call “1g”. Because the strength of the gravitational field follows an inverse-square law (it gets divided by four every time the distance doubles) and the Moon is so much closer to the Earth than it is to the Sun, the Moon is caught in the Earth gravitational catchment area. You may wonder if this means the Moon isn’t orbiting the Sun then. Quite simply, there is no contradiction, it is possible for a celestial object to orbit another object and as part of that 2-body system to also orbit a more massive object. Hence the Moon orbits the Earth, they both revolve around the Sun and the Solar system enjoys a spin around the centre of the Milky Way.

Back to the theory of gravitation. The framework and formulae developed by Newton in the mid-17^th century worked well in most cases (and still do) though some edge cases suggested something might be amiss. Enters none other than Albert Einstein with his ground-breaking general theory of relativity in 1915, 10 years after working out the theory of special relativity which stipulated the constancy of the speed of light, regardless of the motion of the observer (or the light source) and the invariability of the laws of physics in non-accelerating frames of reference, also called inertial frames of reference (refer to Section 6.b for an explanation). From there it was possible to deduct the mass-energy equivalence (briefly explained in Section 7.b) and other phenomena such as time dilation at high velocity or due to the effect of gravity. This led to the introduction of the concept of spacetime, a four-dimensional continuum and, in the theory of general relativity, gravity was incorporated as a geometric property of spacetime, one which can be observed as a curvature produced by the motion and energy of matter. For example, the theory predicted the trajectory of light would be bent by massive cosmological objects, an effect called gravitational lensing in the jargon, and it has proven to be the case.

This clearly changes the nature of the gravitational field from force field to curvature and thus, in modern physics, gravity is treated as a fictitious force, one experienced only by objects with mass or accelerating (or rotating), i.e. in a non-inertial frame of reference, such as the passengers in a plane experience at take-off and unlike the freefalling observer in Einstein’s famous thought experiment. This is not the end of the story most likely though because the spacetime curvature doesn’t sit well with quantum mechanics. Time will tell, what this space (!)

b) Into the deep end

No discussion about spacetime curvature and time dilation would be complete without mention of black holes, this very special region of space time. The logic behind black holes, and the reason behind their catchy name, is that gravity there is so strong that no matter or radiation can escape. This include light so a black hole would be a perfect black.

When we think of gravity in the classical way, as a force exerted on an object with mass, since light is made of photons and those have no mass then they should not be trapped. However, they do have momentum and energy and according to the theory of general relativity, these create curvature so photons do experience gravity. Black holes were predicted well before they could be observed, as one would expect with such a region nothing is emitted from (except the speculative Hawking radiation). It is only in 2015 that a suspected collision of two black holes was detected via their emission of gravitational waves (as described in Section 9.e) and in 2017 that they were imaged, even though they are thought to be common, especially at the centre of galaxies where they are supermassive. The black hole at the centre of our Milky Way galaxy has been named Sagittarius A* and is nothing to worry about for a while, being located at a distance of nearly 27,000 light years away.

Supermassive is actually a technical term and refers to black holes with mass in excess of 1 million times the mass of the Sun. I will go into more details on the life cycle of stars but beyond the threshold of about 1.4 solar masses, named the Chandrasekhar limit, a dead star will undergo gravitational collapse to form a neutron star and if the remnant of matter is above a threshold in the range of 2.2 to 2.9 solar masses, the Tolman–Oppenheimer–Volkoff limit, the nuclear forces and thermal pressure can no longer balance gravity and the body will further collapse into a black hole. To give you a sense of proportions, the radius of a one of the stellar black holes involved in the first gravitational wave detection is about 20km and its mass is approximately 62 times the Sun’s.

Last tidbit. The boundary of spacetime beyond which light and matter “fall” into the black hole is called the event horizon and is so named because beyond it no information about the event can be retrieved. If a body were to approach this horizon, gravitational time dilation would increase to the point where, for an external observer, the body would appear to slow down and slow down, forever, never quite reaching the horizon so it would essentially appear frozen, but not quite. As far as the body itself is concerned, say an infalling observer having safely made it through the superheated accretion disk, it would not experience anything abnormal with respect to the passage of time though it would need to deal with the more pressing issue, or should I say stretching issue, of intense gravitational tidal forces.

c) Gravitational potential energy

Now, back to the more mundane yet still very interesting topic of gravitational potential energy. The Encyclopedia Britannica defines potential energy as “stored energy that depends upon the relative position of various parts of a system”, so the gravitational part of it is the one due to the relative positions within the gravitational field. Since in the classical sense gravity can be thought of as objects attracting each other, the amount of gravitational potential energy stored in a body in relation with the centre of mass of the field is a function of its respective distance to this centre of mass, of the strength of the field, and of the body’s own mass.

One way to describe the potential energy of an object in relation to a specific point in the gravitational field would be, all else being equal, as the amount of work required to move this object from the reference point to its current position. Recalling Newton’s second law of motion and the formula F = m.a, the potential energy of an object on Earth at a distance or height above ground follows the formula PEgrav = m.g where g is the Earth’s gravity with a value of 9.8 m/s².

The nature of potential energy makes it an ideal candidate for storage and the gravitational kind can be observed in many applications and technologies. The best example is probably the building of dams to retain large volumes of water at altitude rather than let them flow down. Not only that but nowadays some of the water is being pumped back up in the upstream reservoir in period of low electricity demand and stored until it is more interesting, economically speaking, to make use of it.

If you have already observed and even wound an old-style mechanical clock, you would have noted the weights are being lifted up and the clock mechanism is being powered by the slow move downward of those weights. Likewise, the potential energy stored by the dam is being harvested as the water falls from its original height. In either case, there is a conversion of energy from potential to kinetic, the two combining to represent the mechanical energy of the system.

d) Mechanical energy

As mentioned in the very first paragraph of this chapter, mechanical energy is not a different form of energy and describes the amount of energy in a system solely due to the motion of the bodies within it (their kinetic energy) and their respective positions. This notion was most likely developed because there is a practical aspect associated to it: the ability to do work. In other words, whether it is in the form of kinetic or potential energy, an object with mechanical energy can cause another object to be displaced. Consequently, mechanical energy can be “harvested” and put to practical use from a human perspective.

Reverting to the dam example, the potential energy stored is converted into kinetic energy as the water flows. This kinetic energy is then used to do work on the blades of a water turbine, forcing them to turn, another form of kinetic energy that is itself converted into electricity by a generator. This should very much remind us of the flow of electrons in a circuit used to create work on the light bulb – this was described in Section 7.e.

A car journey provides another comprehensive illustration of energy conversion and the use of mechanical energy. The fossil fuel in the car engine gets oxidized and the combustion creates pressure used to do work through transmission to the wheels. At the end of the mechanical processes, there is conversion of potential chemical energy into kinetic energy as the car accelerates and if the car is going up a hill some gravitational potential energy is also being stored so that the car can roll back down the hill without firing its engine, converting some of its potential energy into kinetic one. This is very much how a roller coaster works too: initial work is applied to the vehicle carrying passengers to lift it up and the potential gravitational energy then goes through a back-and-forth conversion into kinetic energy as the passenger train loses or gains elevation.

With modern electric cars, including hybrid models, some of their kinetic energy can be converted back into electrical potential energy stored in the car’s battery, especially on downhill sections. This is known as regenerative braking and is a very efficient alternative since otherwise a lot of energy is dissipated as heat at the time of conventional braking (the concept of heat was covered in Section 9.c). Heat need not be wasted though and, in industrial or commercial settings, cogeneration plants are able to channel some of it towards productive use, thereby dramatically improving the efficiency of traditional power generation plants.

Not only heat need not be wasted, but it can also often be central to a design requiring multiple energy conversion steps such as in electricity generation. Thermal power plants, like their name suggests, rely on heat to do mechanical work with high pressure steam driving the blades of a turbine. The most frequent sources of heat are from the firing of coal with high calorific content, natural gas or from nuclear fission. Eventually, I am hoping to tackle all of these in details with a full series dedicated to industrial energy production (conversion would be more accurate), storage and distribution.

There is another type of potential energy, besides the gravitational one, that can be used to do work and therefore falls neatly within the category of mechanical energy: it is called elastic energy. It is stored by work performed on it, generally compression, which at the microscopic level reduces the interatomic distances between the nuclei despite the repulsive forces at play. So it is not a stretch to think about it as another form of potential energy based on the respective positions of atoms and it even has a name: interatomic potential. Note that not all compressions and other types of stresses on a material will result in an elastic deformation characterized by its reversibility. When the deformation is permanent, we instead refer to plasticity.

A classic example of elastic behaviour able to store energy by applying work on it would be coiled springs. This is the mechanism you use when winding a watch and the stored energy is slowly deployed, as the springs uncoil, to power the watch through other types of mechanical energy. If you pull on the string of a bent bow, you would also be storing elastic potential energy and, upon release, this would be transferred into kinetic energy in the arrow as it flies and upon impact (hopefully on the intended target), it would be dissipated as other forms of energy.

e) Friction and material stress

In an isolated system the amount of mechanical energy is conserved, yet if we think back about a car, it will eventually come to a stop if the engine is powered off, even assuming a perfectly flat road and clearly, as we brake, a lot of energy is dissipated via heat, meaning the quantum of energy in the system has decreased. The explanation, of course, is not that energy isn’t fully conserved but that real systems are different from idealized ones and are seldom perfectly isolated. Instead, they are subject to what is termed non-conservative forces such as friction and stress. To be sure, those are not new forces and their properties have all been covered in this series to a greater or lesser degree; they are simply deemed non-conservative as far as the system under study is concerned because they are exiting it so that, to continue operation, the system relies on continuous or ad hoc inward transfers of energy.

The two main types of non-conservative forces are friction and non-elastic material stress. The latter suggests the deformation undergone by the material is at least partially permanent, as in plastic deformation, so the inter-atomic structure will not revert to its original state. With liquids and gases, only to the extent the volume is being compressed are we dealing with elastic energy, otherwise the energy will typically be transferred to the fluid as heat and translate into an increase in temperature or average kinetic energy. In a solid there may be some heating and there is likely to be some deformation as well, a rearranging of the internal structure of the material with higher embedded interatomic potential energy. When the arrow hits the target, some heat will dissipate across the material of the target and there will be some deformation visible in the shape of a hole created by the compression of the material.

As regards friction, it is somewhat of a catch-all term and the easiest way to think about it is as resistance between the surface of two objects moving past each other with sliding generally dissipating more energy by friction compared to rolling. Friction comes under many names but eventually the ultimate reason for energy transfer tends to be the imparting of motion to the atomic structure of the resisting material, thereby increasing either its temperature (average kinetic energy) or the potential energy captured by its inter-atomic arrangement. For instance, at a macro level air drag is created by the compression of air molecules and the friction of car tyres on the road will lead to the deformation and wear of the asperities on the surface of both the road and the tyres. And thus, our roller coaster comes to a standstill and so does a pendulum, having transferred away all their mechanical energy.

f) Trivia – Gravity assist

For this final section, I was initially tempted to discuss the Coriolis force and the Foucault pendulum, an experiment used in the mid-1850s to provide evidence of the Earth’s rotation. Nevertheless I decided this topic may require too much visual content to be practical and enjoyable. Writing more about gravity and orbits, I then toyed with the idea of the three-body problem and Lagrange points, truly interesting subjects but perhaps a little too mathematically-oriented. From there it was only a short hop to the concept of gravity assist, and so here goes.

When launching a space probe, and one day in cases of interstellar travel, there are many aspects to consider and, among those, duration and cost are critical factors. Duration is a function of the route being travelled and the speed of the object along this route with speed itself being predicated on the capacity to accelerate hard, for a long time and then decelerate long enough so as not to overshoot the destination. The more one wishes to accelerate and then decelerate, the more fuel is required if only the gravity of Earth, the Sun and the final destination are considered. The amount of fuel required also increases in proportion with the payload, which includes the cargo, the body of the probe or space shuttle…. and the fuel itself. Just like a plane, the more fuel it requires, the more fuel it needs to accelerate this fuel into the air or escape gravitational pull generally. This obviously drives up cost very significantly.

Fortunately, there is a clever way to save on fuel by using the laws of physics to increase or decrease velocity without fuel burn; it is called gravity assist and works both ways. I will describe the assist for acceleration and it should then become obvious how it can work for deceleration.

Imagine a space probe inbound into Jupiter orbit with the following trajectory that can be traced on a diagram: the Sun is placed at the centre of the reference frame where the axes of the 3 dimensions meet and Jupiter is perfectly aligned on the Y axis, we can call due North at marker 100. Let’s assume the probe incoming speed is 10 per 1 unit of time, there is no atmospheric friction and the manoeuvre occurs on a flat plane, so we can ignore the depth axis Z and concentrate on x.y coordinates. As the probe gets closer to Jupiter, the gravitational pull exerted on it results in an acceleration, an increase in velocity in a straight northern direction if the probe is heading for the centre of the planet – for our purpose we will assume it doesn’t take this suicidal trajectory and flies-by just to the “left” but still describe this as straight north, up the Y axis. Once it has reached the closest point in its orbit around the planet it will start slowing down all the way back to the original speed in relation with Jupiter at the point it exits the planet’s gravitational field. So what’s the deal?

Well, one thing will have happened, because Jupiter both rotates on itself and revolves around the Sun. Let’s further assume the revolution is in a clockwise direction from the position we are looking at and during the period of time when the probe was in orbit the planet has travelled a distance of 7 units exactly along the x axis, that is towards the right on our diagram. As such, when it is ejected out of orbit, the probe still travels at a velocity of 10 away from Jupiter but it has also acquired Jupiter’s velocity in reference to the Sun. If we add both vectors, they form a right triangle with its hypotenuse representing the new velocity in relation with the Sun. The maths are quite simple and follow the Pythagorean theorem: the new velocity V2 is the square root of the sum of the square of the original velocity V1 and of VJ, which represents Jupiter’s velocity: V2 = √(V1² + VJ²) = √(100+49) ≈12.2. This is an increase in velocity of 22%, courtesy of Jupiter. Pretty neat, right?

But what exactly happened here? By now we are sufficiently versed into the laws of physics to know about energy conservation and that there is no free-assist. This obviously points to energy transfer and it would have had to take place during the probe orbiting phase around Jupiter. The answer is simple: gravity is a 2-way interaction so the additional kinetic energy transferred into the probe, according to our frame of reference centred around the Sun, is actually a transfer of Jupiter’s angular momentum and in this process, Jupiter’s angular momentum would have decreased by the same amount of energy – not even in the category of rounding errors as far as the gaseous giant is concerned. If one wanted to decelerate, a flight path designer would need to look for a transfer of angular momentum resulting in a lower Sun-centric velocity. In case you have not read it yet, the concept of angular momentum was developed in Section 8.d.

This type of manoeuvres, known as gravitational assists in the jargon of orbital mechanics, is not just the domain of movies like The Martian and Interstellar (around a black hole, no less), it is used frequently in space exploration. The first instance was by the probe Luna 3 launched in 1959 when the Soviets photographed the far side of the Moon – the side we can’t see because of the tidal lock. Since then, it has famously been used to catapult Voyager 1 and Voyager 2 as fast as possible into interstellar space and the former is currently travelling at 17km/s relative to the Sun; at the date of this writing, among the objects created by man, it also has the distinction of being the farthest from Earth. Very impressive indeed, yet in my opinion the most breath-taking trajectory was that of the MESSENGER probe: launched in 2004, it finally inserted itself into Mercury’s orbit in 2011 after taking the scenic route. Indeed, the main challenge with Mercury is its small size and proximity to the gigantic Sun. This makes staying within Mercury’s orbit rather difficult and if the probed arrived too fast it would have exited and fallen directly into the Sun’s orbit and be captured by it. To decelerate sufficiently without embarking an inordinate amount of fuel, MESSENGER did no less than a total of six flybys across three different planets: Earth (one year after launch), Venus and Mercury itself. Creative thinking would be an understatement.

And… that’s a wrap for Series 1.