Pages

Sunday, 29 January 2012

What is Gravity? (1/2)

We know we don't fall off the Earth, but what is it that holds us on there? And how strong is it? If everyone in China jumped at the same time, what would happen?

Firstly, We think we should start by welcoming all our new followers. As you all will have realised, we post regularly, every Sunday, on a range of topics within maths and physics. These span from relativity and the nature of time to the maths of music. We hope you all will continue to visit the blog and read our posts, so now, lets get started:

Everything we see, hear and feel in the universe are due to four fundamental interactions. I have already talked briefly about them in our post: "What are the Subatomic Particles?" and now I am going to talk about one in particular. Gravity is a force that confused people for hundreds of years, and the only force that has not yet been described by quantum physics. In fact, though people had been studying its effects since the Greeks, the first theories of gravity started being formed around the beginning of the 17th century. Galileo Galilei was questionably the first modern physicist, people had been trying to understand the world and the universe but he was the first to do experiments but then describe and test them with mathematical language.

In his first tests of gravity, he rolled balls down slopes and measured the distance they traveled in a certain amount of time. He realised that if he let the balls roll for twice as long, the distance they travel would increase four times. Because this result was consistent no matter the angle of the slope or the weight of the ball, Galileo said that the same must true for a ball free-falling vertically. He worked out that this meant the balls were accelerating uniformly and then thought about two bodies falling in a vacuum, a very new idea. He came up with a theory that was brand new for the time, objects should all fall at the same speed, the only reason they did not was because of the presence of air. If two objects we dropped in a vacuum, or near vacuum, they would fall at exactly the same speed. He was said to have tried experiments, dropping two balls of the same shape but different masses from a tower and got the result he expected, they fell at the same speed, although these are believed to be thought experiments rather than actually being carried out. However 406 years after his birth, the crew of Apollo 15 went one further, dropping a hammer and feather on the moon. The result was as Galileo predicted:
Galileo went on with his experiments and formulated ideas that later became Newton's first law, the law of inertia and, based on his previous discoveries, theorised that in a frictionless environment, accelerated object's trajectory would be a perfect parabola.

The next big development in gravitation theory came only 45 years after Galileo's death in 1645, it is probably one of the most widely know scientific laws ever, and it was all caused, apparently, by someone getting hit on the head by an apple. It is of course Newton's law of universal gravitation, or more simply, Newton's law of gravity. Newton was very interested in the way that objects moved. This lead him to create his laws of motion. He was in competition with another famous physicist, Robert Hooke, and they were both working on a theory on why planets orbit in an elliptical shape. Edmond Halley came to him asking if there was a inverse square law of attraction to the sun, what orbit would a planet have? He realised that he had found the answer to the question. When writing the theory he found out that there we many things he needed to describe that did not yet have a definition, he was to first physicist to describe what we call mass, as mass, and gave the quantity of motion as the product of velocity and mass, what we now call momentum. This was what lead to Newton's law of universal gravitation. In this case, the inverse square law means that the force that one object imposes on another is proportional to the inverse square of the distance between them. The force is also proportional to the distance between the two objects. The equation that emerges is
Which simplifies to
where m1 and m2 are the different masses of the two objects, and r is the distance between them.
 So Newton's theory states that in a system with two masses in it the force of attraction between them is

Where G is the gravitational constant, it is the value that is used to change the two sides of the equations from being proportional to being equal and the reason we use the symbol instead of the actual number is because it is roughly 6.67300×10-11m3kg-1s-2. So why don't we address the idea that, if everyone in China jumped at the same time, it would shift the orbit of the earth. The two steps we have to go through is to see how much force is exerted on the earth, and then by how much the gravitational force pulls back. Firstly, we are going to assume that the average jump distance for someone to jump is 20cm and the average mass of a person is 70kg and we are going to round the population of China down to 1.3 billion. We are also going ignore the effects of air resistance.

We are going to find out the velocity of a person as they start jumping. This requires a little bit of knowledge of energy, if you do not have this, then you can visit our post on energy. Anyway, when the person jumps they change potential energy to kinetic energy and then back to potential energy. We are just interested in the initial velocity when they jump so we are going to look at the change from kinetic to potential.
Gain in gravitational potential energy = Loss in kinetic energy


So we just found the average velocity of the jump, and we know the distance they jump, so we can work out the time it takes, ds=t so 0.22=t and time=0.1 seconds. From that we can work out that the starting velocity is 4 m/s. Now we can calculate the force that one person jumping exerts on the earth. In this case we are going to have to assume that the amount of time it takes for someone to go from a stand still to moving upwards to be about a twentieth of a second:
Force = Change in momentumTime taken

Where p is momentum
So if all 1.3 billion people in China jump at the same time, the amount of force exerted on the earth will be about 1.82 trillion newtons of force. This may seem like a lot but lets plug that into an equation that will show the acceleration of the earth due to it: 
Force = mass x acceleration
F=ma
1820000000000=5.9722 x 1024 x a
a =0.0000000000003

m is the mass of the earth
So we can already tell that this change is minimal. The earth would accelerate away from China at a speed of 3 x 10-13 meters per second, equivalent to 4.392 x 10-12 kilometres per hour. This movement is negligible seeing as the earth already moves at 29 kilometres per second in its orbit. However, if we are not done yet. Even if this change was large, what is to say that the force of gravity would not just pull the earth, and the people back to their original position? Well this is were Newton's theory comes in. We are not going imagine the people in China as individual people, but instead a large clump of mass.
m1 is the mass of the earth; m2 is the mass of all the people in China; r is the mean radius of the earth plus the 20 cm the people jump.
As you can see, there is a large force pulling them together, if we plug this force into the equation of force to find the acceleration of Earth we get: 
Force = mass x acceleration
As we can see this acceleration, 1.4981 x 10-7 towards the jumping people is actually larger than that away from them, though in the less than half a second they are in the air, the movement of the Earth would be just as negligible. The force when they land would be just as small. When the people fall they gain momentum.
Where x is distance and u is initial velocity
Apply conservation of momentum
So we can see that the change in velocity is also very small.

In conclusion, if everyone in China jumped at the same time, the Earth would accelerate towards the jumping people at a rate of 1.4981 x 10-7 ms2 and would move in the other direction at 4.31 x 10-16 ms when they land, though both of these numbers are very rough estimates made with a lot of assumptions. Despite that, if all those in China were to jump at the same time change in the Earth's speed/orbit would not be noticeable.

In two weeks, we will post again on gravity. We shall move on from Newton's Law of Universal Gravitation on to Einstein's General Theory of Relativity, and then Quantum Gravity. Less on the effects of gravity, and instead what gravity actually is. We have written posts on Einstein's relativity and how it explains that time is not as constant as we think before, if you would like to read those, click hereThe post is up!

Please follow us, @theaftermatter, or email us at contactus@theaftermatter.com. We really like hearing your feedback or just talking about the posts or other physics and maths. We hope you enjoyed this post.

Ned.

Check out our last two posts:
What is Music? - How can music be described mathematically? Why do different instruments sound different? What actually is sound? (If Stephen Fry liked it then we are sure you will!)
What are the Subatomic Particles? - What are the most basic things that make up everything we see, hear and know?

3 comments:

  1. This comment has been removed by a blog administrator.

    ReplyDelete
  2. This comment has been removed by the author.

    ReplyDelete
  3. Hi, I am thankful to you for sharing this awesome article with this helpful knowledge.

    Tim Hoff

    ReplyDelete