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Subsections

# Physical Laws

The mode of inquiry, and the form of reasoning, that physics is based upon can only be grasped by following this process in the actual construction of the leading ideas of physics. The choice of subjects in these lecture notes is intended to both demonstrate the methodology of physics by going through the chain of reasoning that leads to the central ideas in physics, while at the same time explaining in some detail the structure of these leading ideas. Methodology cannot be separated from the content of knowledge, and hence one has to actually reason the reasoning of physics by engaging with the ideas from which this reasoning emerges.

# Domains of Physical Phenomena

To have a meaningful understanding of the vast range of physical phenomena, we need to know at what length scales and time scales these phenomena occur. All physical bodies are characterized by dimensional quantities of which the dimensions of length and time are of fundamental importance. Length is measured in meters and time in seconds. For example, we know that objects that we are familiar with have a size (called scale) of around 1-100 meters. Similarly events that we are witness in daily life have a duration from one second (a heartbeat) to days and years.

Table 2.1: Some typical Length and Distance Scales
 Length (or distance) Meters (approximate) Radius of nucleus Atom A virus Grain of salt Height of a child 1 Distance to the Sun Distance to the nearest star Distance to the center of our Galaxy Distance to the nearest Galaxy Distance to the furthest Galaxy

However, we have no direct experience to decide for example, what is the size of a typical atom. Or how many atoms and molecules constitute a living cell. Or how far are the distant galaxies. To obtain a more complete classification of nature, we need recourse to scientific experiments. As one can imagine, to even arrive at such a classification of physical phenomena, we need to know what discipline studies the phenomena in question.

Table 2.2: Some typical Time Scales
 Time interval Seconds (approximate) Light crosses a nucleus Lifetime of an unstable subnuclear particle Period of an atomic vibration Light travels 1 m Time of one heartbeat 1 One year 3 Typical human life span Age of pyramids Life on earth Age of Universe

Table 2.1 gives a sample of phenomena on different length scales and Table 2.2 does the same for different time scales. Various theories of physics are applicable for different domains of phenomena. All theories of physics are an approximate description of nature, although it is worth noting the remarkable fact that so far no experiment has been able to find any inaccuracy with quantum mechanics. For objects that are not too big or too small, Newtonian, or classical physics is a valid approximation. In the Figure 2.1 we chart out the various length and time scales, or equivalently, length and velocity scales for which the various theories are good approximations.

Figure 1.1 lists all the theories of physics, and as one moves towards the right, the theories encompass a greater range of phenomena and are more accurate as well. The crowning glory of theoretical physics is embodied in string theory. String theory has as yet no experimental evidence in support of its correctness. Nevertheless, it has made a number of theoretical advances, the most important being the successful synthesis of quantum theory with the geometry of spacetime. Einstein's theory of gravitation is incorrect at short distances, and emerges from string theory as an approximation valid at large distances of the more fundamental equations of string theory. A remarkable prediction of string theory is that the physical universe is at least ten dimensional, with only four dimensions being visible to our senses, and the remaining six dimensions existing in the unseen.

# Dimensions and Units

All physical quantities are described by parameters such as length, mass and so on, all of which have dimensions. In general we will denote the dimension of length by , of time by and of mass by . The dimension of a quantity will be denoted by square brackets as . For example, the dimension of the volume of an object is denoted by . An equation such at is consistent only if the dimension of both the sides match, in this case only if all the three quantities and have the same dimensions. Checking that the dimensions agree for an equations is one of the first steps that one should take when analyzing a new set of equations. Dimensional analysis can lead to fairly sophisticated results; however, dimensional analysis can only lead to the general form of the equation, and cannot fully specify the solution. For example, if one has two dimensional quantities and , and we know that the solution is a dimensionless combination of and , then we can straight away conclude that the solution must be a function only of the ratio ; however, based only on dimensional considerations the solution can be either or equivalently , and hence the complete answer for cannot simply be obtained from dimensional analysis. The units that one uses for the various dimensional quantities are specific choices of what is one unit of length, time and so on. Units are arbitrary, and any set of consistent units can be used. For example, we can use meters or equivalently inches to measure length, grams or pounds to measure mass, and so on. Of course one can always convert from one set of units to another, and is analogous to converting from one currency to another. We will use the SI (Systeme International)units, which previously was known as the MKS system of units. In these units distance is measured in meters (m), mass is measured in terms of kilogram (kg), and time is measured in seconds (s). Hence the dimension of velocity is given by , and in SI units we have .

# Experimental Uncertainties

As discussed above, experiments are fundamental to science, and experimental verification of scientific theories is a unique feature of scientific truths. In addition to verifying scientific theories, experiments freely explore nature very much in the way a cook experiments with new recipes. New conditions are created in the laboratory, and the behaviour of nature is then observed. Experiments and observations are carried out using instruments and experimental apparatus. The precision and accuracy of a measurement is determined by the instrument being used. For example, consider cutting a piece of cloth after measuring out a certain length. Suppose the smallest division in the measuring tape is say 0.01m. Since this is the most accurate measurement we can make, our inability to resolve the length of the cloth to any greater degree results is one form of experimental error, called a systematic error. We may measure the cloth to be
 (2.1)

Systematic errors in general result from inherent limitations of the instrument being used. Systematic errors also arise from an incorrect calibration of an instrument, and every measurement made will contain this error. There is another kind of error, called random error.This error arises because the object being measured may not be in exactly the same condition every time we measure it. In the example of the cutting a piece of cloth, the length of the cloth might not be measured in exactly the same manner, resulting in random errors. Random errors can be reduced by repeatedly making the measurement, in the example of the cloth by measuring the length of the piece to be cut. If we measure the length say a 100 times, then the error can be further reduced by measuring the length say 10,000 times and on so.

# Order of Magnitude Estimates

Before embarking on any calculation or experiment, it is a good habit to first do a crude estimate of the magnitude of the result that one expects. Suppose one needs to estimate the mass of water in a water tank which is a cylinder of radius 9m and height 2m. Since we are only interested in an order of magnitude analysis, we can make many approximations. We hence do the following estimate.
 (2.2) (2.3) (2.4)

Given the mass of of water is , we have that the mass of water in the tank is approximately tons. Although this is not an exact answer, we have an estimate of what to expect. Hence it would be disastrous if we designed the tank to hold say ton of water.

# Mode of Presentation

No prior knowledge of physics is required for reading the material presented. A working knowledge of the following mathematics will be helpful.
1. Elementary algebra, in particular the ability to use and manipulate symbols. For example, symbols or and so on normally mean that these two quantities are in some way related, and that is why they have been named in a manner to reflect their connection.
2. Elementary trigonometry, in particular familiarity with sines and cosines.
3. Elementary geometry, in particular, the properties of triangles.
4. Elementary knowledge of vectors. Throughout the text, vectors will always be denoted by boldface, namely a symbol v means that v is a vector and that .
5. Complex numbers are necessary in the study of quantum theory, and we will review the essential properties of these.
6. We will almost always use the symbol to denote points of space and to parametrize time.
Results that are more complicated - such as those derived using knowledge of calculus - or that involve more advanced ideas, as well discussions on scientific methodology, are placed in a box to separate it from the main text. These boxes can be skipped without affecting the reasoning being followed in the main text. All equations not in boxes can be derived without the use of calculus, and all readers are encouraged to do so. We will often have to study small changes in a system when some independent parameter specifying the system undergoes a change. We will denote changes which are finite by the Greek symbol delta, denoted by . So, for example, change in time from instant to will be denoted by . The limit of calculus is to take, for example, to be vanishingly small, that is take . When we do take such a limit, we will place these equations in a box.

Next: Energy Up: Laws of Physics : Previous: Science and the Scientific   Contents
Marakani Srikant 2000-09-11