Fun Facts About Gravity
Gravity of a Five-Body System
Let's look at various examples of gravity that we see in the solar system. We have the Moon orbiting the Earth, and our sphere orbits the Sun (along with the other planets). While the system is always changing, it is, for the most part, a stable one. But (in an orbital system of two similarly massed objects), if a third object of comparable mass enters that system, to put it lightly, it creates chaos. Because of competing gravitational forces, one of the three objects will be ejected and the remaining two will be in a closer orbit than before. Nevertheless, it will be more stable. All of this results from Newton’s Theory of Gravity, which as an equation is F = m1m2G/r^2, or that the force of gravity between two objects equals the gravitational constant times mass of first object times mass of second object divided by distance between objects squared.
It is also a result of the Conservation of Angular Momentum, which simply states that the total angular momentum of a system of bodies must remain conserved (nothing added nor created). Because the new object enters the system, its force on the other two objects will increase the closer it gets (for if distance decreases, then the denominator of the equation decreases, increasing the force). But each object pulls on the other, until one of them has to be forced out to return to a two system orbit. Through this process, angular momentum, or the tendency of the system to continue as is, must be conserved. Since the departing object takes some momentum away, the remaining two objects get closer. Again, that decreases the denominator, increasing the force the two objects feel, hence the higher stability. This entire scenario is known as a “slingshot process” (Barrow 1).
But, what about two two-body systems in near proximity? What would happen if a fifth object entered that system? In 1992, Jeff Xia investigated and discovered a counter-intuitive result of Newton’s gravity. As the diagram indicates, four objects of the same mass are in two separate orbiting systems. Each pair orbits in the opposite direction of the other and are parallel to each other, one above the other. Looking at the net rotation of the system, it would be zero. Now, if a fifth object of a lighter mass were to enter the system in-between the two systems so that it would be perpendicular to their rotation, one system would push it up into the other. Then, that new system would also push it away, back to the first system. That fifth object would go back and forth, oscillating. This will cause the two systems to move away from each other, because the angular momentum has to be conserved. That firth object receives more and more angular momentum as this motion goes on, so the two systems will move further and further away from each other. Thus, this overall group “will expand to infinite size in finite time!” (1)
Strength and Weight
Looking at athletes, many wonder what the limit to their capabilities is. Can a person only grow so much muscle mass? To figure this out, we need to look at proportions. The strength of any object is proportional to the cross-sectional area of it. The example Barrows gives is a breadstick. The thinner a breadstick is, the easier it is to break it but the thicker is the harder it would be to snap it in half (16).
Now all objects have density, or the amount of mass per a given amount of volume. That is, p = m/V. Mass is also related to weight, or the amount of gravitational force a person experiences on an object. That is, weight = mg. So since density is proportional to mass, it is also proportional to weight. Thus, weight is proportional to volume. Because area is square units and volume is cubic units, area cubed is proportional to volume squared, or A3 is proportional to V2 (to get unit agreement). Area is related to strength and volume is related to weight, so strength cubed is proportional to weight squared. Please note that we do not say they are equal but only that they are proportional, so that if one increases then the other increases and vice versa. Thus as you get larger, you do not necessarily get stronger, for proportionally strength does not grow as fast as weight does. The more of you there is, the more your body has to support before breaking like that breadstick. This relation has governed the possible life forms that exist on Earth . So a limit does exist, it all depends on your body geometry (17).
The Shape of a Bridge
Clearly, when you look at the cabling that runs between pylons of a bridge, we can see that they have a round shape to them. Though definitely not circular, are they parabolas? Amazingly, no.
In 1638, Galileo tested out what the possible shape could have been. He used a chain hung between two points for his work. He claimed that gravity was pulling the slack in the chain down to the Earth and that it would have a parabolic shape, or fit the line y2 = Ax. But in 1669, Joachim Jungius was able to prove through rigorous experimentation that this was not true. The chain did not fit this curve (26).
In 1691 Gottfried Leibniz, Christiaan Huygens, David Gregory, Johann Bernoulli finally figure out what the shape is: a catenary. This name derives from the Latin word catena, or “chain.” The shape is also known as a chainette or a funicular curve. Ultimately, the shape was found to result not only from gravity but from the tension of the chain that the weight caused between the points it was attached to. In fact, they found that the weight from any point on the catenary to the bottom of it is proportional to the length from that point to the bottom. So the further down the curve you go, the greater the weight that is being supported (27).
Using calculus, the group assumed that the chain was of “uniform mass per unit length, is perfectly flexible, and has zero thickness” (275). Ultimately, the math spits out that the catenary follows the equation y = B*cosh(x/B) where B = (constant tension)/(weight per unit length) and cosh is called the hyperbolic cosine of the function. The function cosh(x) = ½*(ex + e-x) (27).
A favorite of the Olympics, this event used to be straight forward. One would get a running start, hit the pole into the ground, then holding onto the top launch themselves feet-first over a bar high up in the air.
That changes in 1968 when Dick Fosbury leaps head-first over the bar and arching the back, totally clearing it. This became known as the Fosbury Flop and is the preferred method for pole vaulting (44). So why does this work better than the feet-first method?
It is all about mass being launched to a certain height, or the conversion of kinetic energy to potential energy. Kinetic energy is related to the speed launched and is expressed as KE = ½*m*v2, or one-half mass times the velocity squared. Potential energy is related to the height from the ground and is expressed as PE = mgh, or mass times gravitational acceleration times height. Because PE is converted to KE during a jump, ½*m*v2 = mgh or ½*v2 = gh so v2 = 2gh. Note that this height is not the height of the body but the height of the center of gravity. By curving the body, the center of gravity extends to outside the body and thus gives a jumper a boost they normally would not have. The more you curve, the lower the center of gravity is and thus the higher you can jump (43-4).
How high can you jump? Using the earlier relation ½*v2 = gh, this gives us h = v2 / 2g. So the faster you run the greater the height you can achieve (45). Combine this with moving the center of gravity from inside your body to the outside and you have the ideal formula for pole vaulting.
Designing Roller Coasters
Though some can view these rides with great fear and trepidation, roller coasters have a lot of hard engineering behind them. They have to be designed to ensure maximum safety while allowing for a great time. But did you know that no roller coaster loops are a true circle? Turns out if it were that the g forces experience would have the potential to kill you (134). Instead, loops are circular and have a special shape. To find this shape, we need to look at the physics involved, and gravity plays a big role.
Imagine a roller coaster hill that is about to end and drop you off into a circular loop. This hill is a height h tall, the car you are in has mass M and the loop before you has max radius r. Also note that you start higher than the loop, so h > r. From before, v2 = 2gh so v = (2gh)1/2. Now, for a person at the top of the hill all the PE is present and none of it has been converted to KE, so PEtop = mgh and KEtop= 0. Once at the bottom, that entire PE has been converted to KE, to PEbottom = 0 and KEbottom = ½*m*(vbottom)2. So PEtop = KEbottom. Now, if the loop has a radius of r, then if you are at the top of that loop then you are at a height of 2r. So KEtop loop = 0 and PEtop loop = mgh = mg(2r) = 2mgr. Once at the top of the loop, some of the energy is potential and some is kinetic. Therefore, the total energy once at the top of the loop is mgh + (1/2)mv2 = 2mgr + (1/2)m(vtop)2. Now, since energy can neither be created nor destroyed, the energy must be conserved, so the energy at the bottom of the hill must equal the energy at the top of the hill, or mgh = 2mgr + (1/2)m(vtop)2 so gh = 2gr + (1/2)(vtop)2 (134, 140).
Now, for a person sitting in the car, they will feel several forces acting on them. The net force they feel as they ride the coaster is the force of gravity pulling you down and the force the coaster pushes up on you. So FNet = Fmotion (up) + Fweight (down) = Fm – Fw = Ma - Mg (or mass times acceleration of car minus mass times acceleration of gravity) = M((vtop)2)/r – Mg. To help make sure that the person will not fall out of the car, the only thing that would pull him out would be gravity. Thus the acceleration of the car must be greater than the gravitational acceleration or a > g which means ((vtop)2)/r > g so (vtop)2 > gr. Plugging this back into the equation gh = 2gr + (1/2)(vtop)2 means gh > 2gr + ½(gr) = 2.5 gr so h > 2.5r. So, if you want to reach the top of the loop courtesy of gravity alone, you much start from a height greater than 2.5 times the radius (141).
But since v2 = 2gh, (vbottom)2 > 2g(2.5r) = 5gr. Also, at the bottom of the loop, the net force will be the downward motion and the gravity pulling you down, so FNet = -Ma-Mg = -(Ma+Mg) = -((M(vbottom)2/r + Mg). Plugging in for v bottom, ((M(vbottom)2)/r + Mg) > M(5gr)/r + Mg = 6Mg. So when you get to the bottom of the hill, you will experience 6 g’s of force! 2 is enough to knock out a kid and 4 will get an adult. So how can a roller coaster work? (141).
The key is in the equation for circular acceleration, or a c = v2 / r. This implies that as the radius increases, the acceleration decreases. But that circular acceleration is what holds us to our seat as we go across the loop. Without it, we would fall out. So the key then is to have a large radius on the bottom of the loop but a small radius on the top. To do this, it must be taller than it is wider. The resulting shape is what is known as a clothoid, or a loop where the curvature decreases as the distance along the curve increases (141-2)
Running vs. Walking
According to official rules, walking is different from running by always maintaining at least one foot on the ground at all times and also keeping your leg straight as you push off the ground (146). Definitely not the same, and definitely not as fast. We constantly see runners breaking new records for speed, but is there a limit to how fast a person can walk?
For a person with leg length L, from sole of foot to the hip, that leg moves in a circular fashion with the pivot point being the hip. Using the circular acceleration equation, a = (v2)/L. Because we never conquer gravity as we walk, the acceleration of walking is less than the acceleration of gravity, or a < g so (v2)/L < g. Solving for v gives us v < (Lg)1/2. This means that the top speed a person can reach is dependent on the leg size. The average leg size is 0.9 meters, and using a value of g = 10 m/s2, we get a v max of about 3 m/s (146).
Eclipses and Space-Time
In May 1905, Einstein published his special theory of relativity. This work demonstrated, amongst other work, that if an object has sufficient gravity then it can have an observable bending of space-time or the fabric of the universe. Einstein knew that it would be a hard test, because gravity is the weakest force when it comes to small-scale. It would not be until May 29th, 1919 that someone came up with that observable evidence to prove Einstein was right. Their tool of proof? A solar eclipse (Berman 30).
During an eclipse, the Sun’s light is blocked out by the Moon. Any light that comes from a star behind the Sun will have its path bent during its pass near the Sun, and with the Moon blocking out the Sun’s light, the ability to see the starlight would be easier. The first attempt came in 1912 when a team went to Brazil, but rain made the event unviewable. It ended up being a blessing because Einstein made some incorrect calculations and the Brazilian team would have looked in the wrong place. In 1914, a Russian team was going to try for it but the outbreak of World War I put any such plans on hold. Finally, in 1919 two expeditions are underway. One goes to Brazil again while the other goes to an island off the coast of West Africa. They both got positive results, but barely. The overall deflection of the starlight was “about the width of a quarter viewed from two miles away (30).
An even harder test for special relativity is not only the bending of space but also time. It can be slowed down to an appreciable level if enough gravity exists. In 1971, two atomic clocks were flown up to two different altitudes. The clock closer to the Earth did end up running slower than the clock at the higher altitude (30).
Let’s face it: we need gravity to exist, but it has some of the strangest influences we have ever encountered in our lives and in the most unexpected ways.
Barrow, John D. 100 Essential Things You Didn't Know You Didn't Know: Math Explains Your World. New York: W.W. Norton &, 2009. Print.
Berman, Bob. “A Twisted Anniversary.” Discover May 2005: 30. Print.
© 2014 Leonard Kelley