Graphs, errors, significant figures, dimensions and unitsThis page supports the Physclips project, especially Chapter 2: Projectiles
The first chapter in Physclips mechanics uses displacement-time and velocity-time graphs for a man walking in a straight line, so we'll begin with this animation.
An example: Displacement-time graphs
The reference for displacement is the wall, x = 0. We also need a reference for time. It could be the time at which we set a stop-watch ticking. If the watch starts at zero seconds, any time after that is positive time (t > 0). Of course physics was still happening before we set our watch, so anything that happens before t = 0 would be represented on the negative part of the time axis.
Units on graphs
The method used here is to plot x/m and t/s. This has the advantage that, when x is divided by a metre or t is divided by a second, the result is a number. Numbers (not quantities) are what we plot on the axes: the axes really are x/m and t/s, so it is a good idea to to label them in this way.
Units and dimensions
- "The game lasted 15 kg"? Or:
- "The displacement is given by the weight divided by the volume squared"?
We shall see that this very simple idea can be quite important – and also very useful.
Now the units on either side of an equation need not be the same. For instance, I may write
Let's look at more interesting examples. When we write
(Note the use of the negative exponent here. Acceleration has units of metres per second per second, i.e. metres divided by seconds twice, or divided by seconds squared. So we write metres per second per second as m.s−2.)
In the units of the Système International, almost universally used in science, there are no conversion factors for the base units, so we can relate the newton, the unit of force, to other base units:
Let's now see how the method of dimensions can be useful, via this
Example: how does the frequency, of a pendulum depend on the length?
We know that this depends on the length, L – a long one swings more slowly than does a short one. It also depends on the strength g of the gravitational field – it won't swing at all without one. Does it also depend on the mass, m? On the temperature, T? Let's write for the frequency, f:
The other two equations tell us that b = 1/2, and that a + b = 0, so a = −a = −1/2. So, substituting in our original equation for the frequency,
* I raise a couple of tiny caveats, to preempt the pedants. For a pendulum whose mass is comparable with the that of the planet upon which it is mounted, the pendulum mass does appear -- or at least the ratio of these two masses appears. Further, we have cheated a little on the temperature, because we could write temperature in units of energy. Doing so, the conversion factor would be Boltzmann's constant, whose very small size would give us the clue that temperature is only relevant in mechanics for objects of molecular size. And on this molecular scale we should often need to use quantum mechnics rather than Newtonian mechanics.
Physclips is a scientific presentation, and we use only the SI. If you enounter problems stated in other units, the simplest procedure is often to translate the problem into SI, solve it, then translate the back. This sounds like extra work, but it is usually much less than the extra work required in using the imperial system of units, which has internal conversion factors.
In the United States of America, Liberia and Myanmar, the British imperial system is the official system. This system used to be much more widespread, and vestiges of it remain in other countries that are in the process of 'going metric', ie converting to the SI.
Dealing with or converting from the imperial system usually involves just a multiplictaive factor. For instance, the inch, an imperial unit of length, is officially defined to be equal to 25.4 mm. These multiplications can become awkward in some cases: consider this imperial unit of thermal conductivity, one British Thermal Unit per second per square foot per degree Farenheit per inch. One can see why it exists, but it is ugly and inconvenient. (For comparison, the SI unit thermal conductivity is W.m−1.K−1.)
Some confusion arises, however, because of the different colloquial use of units of mass and force in the SI and imperial system. In the imperial system, the unit of force is the pound-force, or sometimes, as in many American physics text books, just the pound. The unit of mass in the imperial system is the slug, which is a mass that is accelerated by one pound at one foot per second per second. The slug is 14.5939 kg. These equations, which are definitions, allow us to compare the units of mass and force:
The slug is very rarely used. Pound is used colloquially as a unit of quantity – a pound of apples colloquially means a quantity of apples that weighs a pound-force (at the earth's surface). There is another imperial unit of force, the poundal. This is defined as the force required to accelerate at one foot.second−2 a mass whose weight is one pound. So a pound is 32 poundals.
The units mentioned above are related to features of the earth (its circumference originally determined the metre, and the second is related to the day) or of artifacts on earth, such as the standard kilogram, or of particular substances, especially water. In contrast, the laws of physics and combinations of them yield natural units, which are used by some theoretical physicists, especially cosmologists. The speed of light, for instance, is taken as the unit for speed. Although this makes equations look simple these units are, in general, inconvenient for measurement. For instance, the natural units of length and time are inconveniently small (The Planck length is 1.6 x 10-35 metres, the Planck time is 5.4 x 10-44 seconds). See The Planck scale for more detail.