Gravitation field of planets. Starships taking off into space. Black holes.
Gravitation field has been the stuff of science fiction ever since Einstein and his theory of relativity.
In school physics, we are interested in something more down to Earth. Or to be more precise (or accurate??), something closer to Earth.
We learn that the weight of an object is really an attraction between the object and Earth. This is an attraction between two masses.
Earth's attraction? masses ??
This is probably the point where it starts to get abstract - or hard to understand for people new to physics.
There are often two main "mental block" issues for students who are new to this topic.
One is the difference between the idea of mass and weight. The other is the idea that one ordinary object - not electrical, not magnetic - can somehow attract each other.
I once tried to explain to a 10 year old student how an object with a mass of 1 kg would still have a mass of 1 kg on the moon because the mass is about the matter, or whatever stuff it is inside the object. And how its weight on the moon would be a lot smaller than its weight on Earth, because weight is about how much the Earth pulls on the object.
It was impossible for him to get the idea. The trouble is that we are so used to calling 1 kg the weight of an object. And we can clearly feel how heavy it is by holding it in our hands. So familiar that it becomes very hard to unlearn this connection and realise that there are two different "physics" properties (or features) about the rice.
One is the "matter" (a more scientific word than "whatever stuff it is") inside the rice - the grains, or the atoms inside the grains.
The other is the weight, a force from the Earth that is pulling at the bag of rice.
If we take the sack of rice to the moon, the rice is still the same rice. It looks the same. It tastes the same if we can cook it on the moon. So the matter (whatever stuff it is) inside the rice grains have not changed. It still tastes the same if we eat it. So it has the same amount of "matter".
But the moon pulls a grain of the rice with a much smaller force than on Earth. This force gives the grain its weight on the moon. So the same sack of rice is a lot lighter on the moon.
I hope the above example does not sound like a circular argument. But through talking to students, I know that it can be psychologically hard to accept this way of thinking once a prejudice (i.e. a preconceived opinion) is formed.
Even for a 10 year old kid that I once talked to !
There is actually some similarity between electricity and gravity. For a student learning this for the first time, this sounds crazy. Electricity gives us all the modern conveniences - lighting, refrigerator, mobile phones, internet ...
What has electricty got to do with gravity?
Historically, Issac Newton was given the credit for coming up with a lot of understanding on gravity.
Around 10 years after Newton died, Charles Coulomb was born. Coulomb made discoveries on what we call "static electricity" today.
It is hard to observe this in Singapore, but when I studied at Cambridge in England, I often got a small electric shock when I put on my shirt, or when I touched the wooden cupboard.
This is because electricity can accumulate on objects through contact or rubbing. As this electricity stays on the object, it is called static electricity. Today, we may just cal it "static".
In Singapore, we seldom have such experience because the humid air can carry away this electricity.
When students learn electricity in physics at school, they would usually start by learning the topic of "static electricity". I shall talk about this in another blog.
Static electricity is similar to gravity in some ways, which is why they like to compare them in a physics course. (Maybe we should do this after learning static electricity?)
But confusingly, to compare these 2 types of forces, we actually need to use something about static electricty that is not present in gravity.
In England, when I comb my hair with a plastic comb, my hair stands up towards the comb! It turns out the tiny electrons jump off the comb onto my hair. The atoms on the comb is then a bit short of electrons.
We describe this by saying that my hair becomes negatively charged, and the comb positively charged.
These two charges can attract each other, like gravity.
And like gravity, the attraction decreases as the comb gets further from my hair. Just as the weight of a ball decreases if it is taken by a rocket further and further from Earth into outer space.
There is a story that Isaac Newton once sat under an apple tree. An apple fell on his head. That was how he discovered gravity. Whether this really happened, it has become the stuff of a legend.
What we do know is that Newton developed a mathematics called calculus - to the extent that he could use it to calculate accurately the oval shaped orbits of planets going arould the sun.
Through this "exercise", Newton did two things - he invented calculus, and he discovered the law of gravitation. Calculus is a type of mathematics that secondary students have a hard time learning.
The law of gravitation states that the force of attraction between two objects is directly proportional to their masses m1 and m2, and inversely proportional to the square of the distance between them.
That is a mouthful of mathematical terms. In English, it means for example that
- if we double the mass of one object, the attractive force between them doubles; - if we double the distance between the two objects, the force drops by 4 times.
In school physics, we learn that the weight of a stone is really the attraction between Earth and the stone. And this attraction is between the mass of Earth and the mass of the stone. This attraction is also what we call "weight" of the stone.
We learn that any two things that have mass would have such a force between them. But unless one or both are as massive as Earth, we cannot feel them. Like if I hold a ball on my left hand and a ball on my right hand, I cannot feel them attracting each other. For such "down to Earth" objects like balls, their gravitational attract for each other is tiny.
So to most people, the idea of gravitational attraction between two masses can still sound like science fiction. Do we really have to imagine stars and planets to understand this force?
In 1798, one person decided to measure the gravitational force between 2 balls. Henry Cavendish knew this force is really weak between between Earthly objects. So he found the densest material he could lay his hand on.
He found two lead balls. Lead objects tend to be very heavy, because it has high density. Gold is denser, but probably a bit expensive for Henry. His setup was probably quite simple :
- a beam balance (just a rod balanced on a string) free to rotate, with with 2 lead balls balanced on it,
- and a bigger lead ball near to each of them.
When he let go of the balance - guess what - the smaller lead balls started moving toward the bigger lead balls !
Now lead is not magnetic, and he did not put any electric charges on them. So it must be gravity - gravitational force between the lead balls !
For the first time in history, we can see gravitational attraction between two Earthly objects !
Using his measurement, Cavendish finally calculated the value of G in the formula :
and caused headache to every generation of high school physics students after him.
Remember the formula
or
where g is the constant 9.81 m/s2 ?
We now have to learn that this is only accurate for objects near Earth's surface. Near means like a few km. As we go higher, g decreases. For example, at 1000 km high, g drops to about 7.3 m/s2.
We would feel lighter definitely. A 1 kg would feel more like a 0.7 kg. Objects also fall a bit slower.
In the topic on Newton's laws, we learn that
When we apply this to gravity, we have
If we divide both sides by mass, we get
This gives us another way to think of acceleration due to gravity - it is also called gravitational field strength.
This way of thinking is useful for solving certain problems.
With all these new terms, it is easy to get confused. So we have tp remember that "gravitational field strength" is really equal to "acceleration of free fall".
Finally, we look at the part of this topic that can cause some headache - the idea of gravitional potential energy.
Life used to be easier when we are grounded on Earth :
where g is acceleration due to gravity.
But when we go to large distances from Earth where "weight" and "acceleration due to gravity" changes a lot, this formula for work done is not valid anymore.
Not only that - there is also the question of where on Earth (literally) do we bring the object to that height? Surely, I would do less work if I lift the object from Mount Everest.
We need a new way to think about the potential energy. What is the scientists' answer?
We bring the object from infinity.
???
If you learn this for the first time, it is difficult even to imagine what it means.
But imagine we have to. Funnily for a subject like physics that studies the real physical world, there is so much imagining needed.
To make it easier to understand, here is how we do it. Suppose we want to find the potential energy of a 1 kg mass that is 10,000 km from the centre of the Earth. This is about 3,600 km above Earth's surface.
This is the idea normally used in physics :
1. Imagine that the object is at infinity. In practice, this means it is so far that it feel practically no gravitational force from Earth.
2. Calculate the work done to bring it to the point that is 10,000 km from the centre of Earth.
The answer is the potential energy of the object, following the new meaning.
There are just 2 small problems in even understanding this scheme:
- First : Infinity? Where is that?
- Second : Do work to bring it to Earth? But Earth's gravity is pulling at it. Why do we need to do any work at all?
Both valid questions. So here is where we have to be flexible - but in a logical way. This is how we calculate the potential energy in practice :
1. Start from the point 10,000 km from the centre of the Earth.
2. Bring the object far from Earth - so far that the amount of work done hardly changes any more.
3. Add a negative sign to the value of the work done. That is our gravitational potential energy, according to this new meaning.
Perfectly logical.
Except that we have a new problem. A negative value for the potential energy? What does that mean? How can energy be negative.
Here is where we have to keep an open mind. The important thing is not to think this is wrong because we have not heard about it before. Instead, we have to focus on two things :
1. is it logical?
2. does it work?
And the answer is yes to both. This understanding has literally taken humans to the moon.
In another blog, I have talked about circular motion. I have explained how a force is need to keep an object in circular motion, even if there is no friction. The simple reason is that if there is no force at all (so no fricion also), an object would just go in a straight line.
Lets think about a satellite going round the Earth. The satellite does not need any fuel to push it up and keep it at that height. But if so, why does it not fall down to Earth ??
There are 2 ways to think about this :
1. Suppose I throw a ball in a horizontal direction. The ball flies off from my hand in a curved path, falls lower and lower, then hit the ground. But what if it can fly so fast that the curved path follows the curve of Earth's surface. Then it would never hit the ground.
2. We have seen in the blog on circular motion that for an object to go round in a circle, it needs a force at right angle to its velocity. This is exactly the case for a satellite. It is sent up to a particular height and given a horizontal velocity. The gravity provides the force that keeps it in circular motion - that centripetal force..
Once the stuff of science fiction, satellites are now very much apart of everyday life.
If you are lost or if you want to know where your loved one is, just flick on you smart phone and bring up the GPS app. North Korea just launched it first spy satellite. Elon Musk helped Ukraine war with his Starlink satellites.
In this syllabus, we are interested in the geostationary satellite. This is a satellite that follows Earth's rotation in such a way that it is always roughly above the same spot on Earth.
For it to remain above the same spot, it has to be directly above the equator and follow the Earth's rotation. A geostationary orbit is a circular orbit 35,786 km above Earth's equator, 42,164 km in radius, and follows the direction of Earth's rotation.
The main uses are communications, weather forecasting and navigation.
You can learn these concepts and more at Dr Hock's maths and physics tuition.