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Posts Tagged ‘Mechanics’

This weekend I was helping my youngest daughter revise for her Christmas physics exam. She tells me she enjoys physics, but I’m not sure whether she’s just saying this to please me! Her brother has just started his physics degree this last September, and her elder sister seriously thought of doing physics for A-level before deciding against it; but I suspect my youngest is more on the languages and creative arts side than a scientist. We shall see, she is only 13.

The material we were going over was the basics of motion, or mechanics. She has been learning about forces, pressure, velocity and resistances to motion (friction and air resistance). I asked her if she had been learning Newton’s three laws of motion, and from her answer I wasn’t sure whether she had or not!

I distinctly remember my own first encounter with Newton’s three laws of motion. I was about my daughter’s present age, and due to a Horizon programme about cosmology and particle physics (that I mention in this blog), I had already decided I wanted to study astronomy and physics. It therefore came as a bit of a shock to me when we were presented with Newton’s laws of motion, and I found I could not remember them.

Despite seeming to have a very good ability to remember poems and song lyrics, I am terrible at remembering ‘facts’, and our physics teacher presented Newton’s three laws to us as a series of facts. After several days of trying and failing to remember them in the words he had used (or which the text book had used), I was beginning to have serious doubts that I could go into physics at all.

And then I had an epiphany. I realised that if I understood Newton’s laws of motion, I did not need to remember them. If I could understand them, I could just state them in my own words; and within less than an hour I felt I had understood them thoroughly (although I’d like to think I have a deeper understanding of them now than I did at 13!). I often say to my students that the only things they need to remember in physics are the things they do not understand. That, certainly, has been my own experience.

Inertia

The concept of inertia is fundamental to our ideas of motion, and yet it is not the easiest concept to understand or explain. But, I will have a go! Galileo was the first person to think of the concept of what we now call ‘inertia’. He realised that a stationary object wants to stay stationary, and you have to do something to it (push it, pull it, or drop it) to get it moving.



Galileo was the first person to outline the concept of inertia.

Galileo was the first person to outline the concept of inertia.



He also realised that objects which are moving want to carry on moving. This is not obvious, as we all know if we give an object a push it may start to move but will slow down and stop. Galileo realised that objects which were moving stopped because of resistive forces like friction or air resistance, and in the absence of these an object would carry on moving. This is, essentially, the idea of inertia. The tendency a body has to remain at rest or to carry on moving.

Newton’s 1st law of motion

Newton’s 1st law of motion is essentially a statement of the concept of inertia, and is sometimes called the ‘law of inertia’. If someone asks me to state Newton’s 1st law the wording I use will probably change slightly each time, but the key idea I make sure I try to get across is the concept of inertia. So, a way to state Newton’s 1st law is


A body will maintain a constant velocity unless a force acts upon it

This is, more or less, the most succinct way I can express his 1st law. But, by stating it so succinctly, there are hidden complications. The first is to realise that the term ‘velocity’ has a very precise meaning in physics. I was trying to explain this to my daughter over the weekend (except we were doing it in Welsh, so using the Welsh word ‘buanedd’).

For a physicist, velocity is not the same as speed, even though we may use them interchangeably in everyday language. Speed (‘cyflymder’ in Welsh ūüėõ ) is a measurement of how quickly an object is moving, but velocity also includes the object’s direction of motion. A stationary object has zero velocity.

Therefore, a more long-winded (but no less correct) way to state Newton’s 1st law is

A body will remain at rest, or carry on moving with a constant speed in a straight line, unless acted upon by a force


The last thing I should mention in relation to the wording of Newton’s 1st law is that, strictly speaking, I should say ‘an external resultant force’, because an object can maintain a constant velocity with more than one force act upon it, as long as those forces are equal in size and opposite in direction. A good every-day example of this is driving a car at a constant speed in a straight line. The car does not continue to do this if we take our foot off of the accelerator, because air resistance and friction will slow the car down. When it is moving at a constant velocity the resistive forces are balanced by the force the engine is transferring to the tyres on the road. An ice puck, once pushed, will take a long time to slow down. This is because the friction on ice is much less than on a rougher surface, so the puck comes closer to acting like Newton’s 1st law says objects should act.

I don’t think my daughter has learnt about Newton’s 2nd law yet, which I often tell my students contains the most important equation in physics. His second law tells us what happens to an object if there is a (resultant external) force acting up on it. I will blog about that next week.

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Anyone who has studied mechanics / dynamics will have come across Newton’s equations of motion (not to be confused with his laws of motion). The ones I get my students to use are


  1. v = u + at \; \; \text{  (Equ. 1)}
  2. s = ut + \frac{1}{2} at^{2} \; \; \text{  (Equ. 2)}
  3. v^{2} = u^{2} + 2 as \; \; \text{  (Equ. 3)}


where u is the initial velocity, v is the velocity at time t, s is the displacement and a is the acceleration. Note, these equations are only true for constant acceleration, but that actually covers quite a lot of situations. They can all be derived from the definition of acceleration.



Newton's equations of motion can  be derived from his 2nd law  of motion.

Newton’s equations of motion can be derived from his 2nd law of motion.



Derivation of Equation 1

We start off with out definition of acceleration, which is the rate of change of velocity. Writing that mathematically,



a = \frac{dv}{dt}

This is an example of a first order differential equation. To solve it we integrate. So we have


a \int dt = \int dv


When we integrate without limits, we have to include a constant term, so we can write


at = v + C


where C is our constant. To determine the value of the constant we need to put in some conditions, such as (but not necessarily) initial conditions. When t=0 we have defined that v=u, so we can write


0 = u + C \rightarrow C=-u \rightarrow \boxed{v = u + at \; \; \text{ (Equ. 1)} }

Derivation of Equation 2

To derive equation two, which we notice involves the displacement (the vector equivalent of distance), we do the following



v = \frac{ds}{dt} = u + at \rightarrow ds = \int u dt + \int at dt
s = ut + \frac{at^{2}}{2} + C


When t=0 \text{ then } s=0 so we can write


0 = 0 + 0 +C \rightarrow C=0 \rightarrow \boxed{s = ut + \frac{1}{2}at^{2} \; \; \text{ (Equ. 2)} }

Derivation of Equation 3

To derive equation 3 we use the trick of writing the acceleration a in terms of the velocity v and the displacement s. To do this we write



a = \frac{dv}{dt} = \frac{dv}{ds} \cdot \frac{ds}{dt} = \frac{dv}{ds} \cdot v = v \frac{dv}{ds}


So, writing


v \frac{dv}{ds} = a \rightarrow \int v dv = \int a ds \rightarrow \frac{v^{2}}{2} = as + C
Again, we can work out C by remembering that s=0 \text{ when } t=0 so


\frac{u^{2}}{2} = 0 + C \rightarrow C = \frac{u^{2}}{2}


and so


\frac{ v^{2} }{2} = as + \frac{ u^{2} }{2} \rightarrow \boxed{ v^{2} = u^{2} + 2 as \; \; \text{ (Equ. 3)} }

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At number 5 in The Guardian’sten best physicists” is the Scottish theoretical physicist James Clerk Maxwell.

 

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As the caption above says, James Clerk Maxwell is probably one of the least known of the physicists in this list, and yet one of the most important. It was through his theoretical work that a full understanding of electromagnetism became possible. His four equations, known as Maxwell’s equations, are as important to understanding electromagnetism as Newton’s laws of motion are to understanding mechanics. In some ways, Maxwell was to Faraday what Newton was to Galileo. Neither Galileo nor Faraday had the mathematical ability to write in mathematics the experimental results they obtained in mechanics and electricity and magnetism respectively. Newton wrote the mathematical equations to explain Galileo’s experiments in mechanics; and similarly Maxwell wrote the mathematical equations to explain the experiments Faraday had been doing on electricity and magnetism.

Maxwell’s brief biography

James Clerk was born in 1831 in Edinburgh, Scotland. He was born into a wealthy family, and later added Maxwell to his name when he inherited an estate as part of his wider family’s wealth. His mother died when Maxwell was eight years old, and at ten years of age his father sent him to the prestigious Edinburgh Academy.

At the age of 16, in 1847, Maxwell left Edinburgh Academy and started attending classes at Edinburgh University. His ability in mathematics quickly became apparent. At the age of only 18 he contributed two papers to the Transactions of the Royal Society of Edinburgh, and in 1850, at the age of 19, he transferred from Edinburgh University to Cambridge. Initially he attended Peterhouse College, but transferred to the richer and better known Trinity College. He graduated from Trinity College Cambridge in 1854 with a degree in mathematics. He graduated second in his class.

In 1855 he was made a fellow of Trinity College, allowing him to get on with his research. But in November 1856 he left Cambridge, being appointed Professor of Natural Philosophy at the University of Aberdeen in Scotland. He was 25 years of age, and head of the Natural Philosophy department, some 15 years younger than any of his colleagues there.

Maxwell left Aberdeen in 1860 to become Professor of Natural Philosophy at Kings College, University of London. He resigned in 1865 to return to his estate in Scotland, and for the next 6 years he got on with his research, and due to being independently wealthy he didn’t need a formal paying position at a university. But in 1871 he was tempted back into university life, he was appointed the first Cavendish Professor of Physics at Cambridge University, and given the role of establishing the Cavendish Laboratory, which opened in 1875.

Maxwell died in 1879 of abdominal cancer, and is buried in Galloway, Scotland, near where he grew up.

Maxwell’s scientific legacy

Maxwell is best known amongst physicists for two main areas of work, his work on electromagnetism and his work on the kinetic theory of gases. As I mentioned above, his work on electromagnetism can be likened to a certain extent to Newton’s work on mechanics. Newton was able to take the experimental results Galileo had found on the behaviour of moving bodies, and find the underlying laws of mechanics to explain them, and to express these in mathematical form. This is essentially what Maxwell did for the work of Faraday, he was able to find the laws which explained the observations Faraday had been making, and was a sufficiently skilled mathematician to write these laws in equation form.

Maxwell first presented his thoughts on electromagnetism in a paper in 1855, when he was 24 years old. In this he reduced all the known knowledge of electromagnetism as it existed in 1855 into a set of differential equations with 20 equations and 20 variables. In 1862 he showed that the speed of propagation of an electromagnetic field was approximately the same as the speed of light, something Maxwell thought was unlikely to be a coincidence. This was the first indication that light was a form of electromagnetism, but a form to which are eyes are senstive.

In 1873 Maxwell reduced the 20 differential equations of his 1855 paper into the four differential equations which are familiar to every undergraduate physics student. These equations, expressed in their differential form are
\boxed{ \begin{array}{lcll} \nabla \cdot \vec{D} & = & \rho & (1) \\ & & & \\ \nabla \cdot \vec{B} & = & 0 & (2) \\ & & & \\ \nabla \times \vec{E} & = & - \frac{\partial \vec{B}}{\partial t} & (3) \\ & & & \\ \nabla \times \vec{H} & = & - \frac{\partial \vec{D}}{\partial t} + \vec{J} & (4) \end{array} }

 

The symbol \nabla is known as the vector differential operator, or del, and I have explained del in this blog.

The four equations can also be written in integral form, which many people find easier to understand. In integral form, the equations become

 

\boxed{ \begin{array}{lcll} \iint_{\partial \Omega} \vec{D} \cdot d\vec{S}& = & Q_{f}(V) & (5) \\ & & & \\ \iint_{\partial \Omega} \vec{B} \cdot d\vec{S} & = & 0 & (6) \\ & & & \\ \oint_{\partial \Sigma} \vec{E} \cdot d\vec{\l} & = - & \iint_{\Sigma} \frac{\partial \vec{B} }{\partial t} \cdot d\vec{S} & (7) \\ & & & \\ \oint_{\partial \Sigma} \vec{H} \cdot d\vec{l} & = & I_{f} + \iint_{\Sigma} \frac{\partial \vec{D} }{\partial t} \cdot d\vec{S} & (8) \end{array} }

Kinetic theory of gases

Maxwell also did important work on one of the other main areas of physics research in the 19th Century – the kinetic theory of gases, a branch of thermodynamics. Maxwell and the Austrian physicist Ludwig Boltzmann independently came up with an expression which describes the distribution of speeds of the atoms or molecules in a gas. The speed depends on the temperature, the higher the temperature the faster the atoms or molecules move. But Maxwell and Boltzmann showed that not all the atoms will be moving at this speed; some will be moving faster and some will be moving slower. The distribution of speeds is known as the Maxwell-Boltzmann distribution.

The theory of colour vision

Less well known to most physicists is the work Maxwell did on optics, and on colour vision. He was the first person to show how a colour image could be created from combining three images taken through red, green and blue filters. This lies at the foundation of practical colour photography, and is also how our colour television sets work.

 

I think it is true to say that most physicists would place Maxwell in their top four physicists. His contributions to physics are immense, creating the formal framework for our understanding of electromagnetism, without which the modern world would not be possible. That he is so unknown outside of the world of physics is strange, but whereas Newton had his F=ma and Einstein his E=mc^{2}, Maxwell’s equations are far too complicated to have permeated into the consciousness of the public. But they are every bit as important as Newton and Einstein’s.

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You can read more about James Clerk Maxwell¬†and the other physicists in this “10 best” list in our book¬†10 Physicists Who Transformed Our Understanding of the Universe.¬†Click here for more details¬†and to read some reviews.

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Ten Physicists Who Transformed Our Understanding of Reality is available now. Follow this link to order

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