Bonus Material For Advanced Learners

• Problem of the Day

How does the value of $\sin^{-1} (0.1 \; \textrm{cm})$ differ from the value of $\sin^{-1} (0.1 \; \textrm{m})$?

• significant figures and error propagation

The guidelines given in our textbook (and in lecture) for dealing with significant figures (also known as significant digits) are fine for an introductory study. However, if you will become a scientist or engineer some day, then you will benefit by learning more careful methods for dealing with experimental uncertainty. The traditional term for this advanced method is "error propagation," but no error has been made. It's just that virtually no scientific observation or measurement can be precise, so all contain some uncertainty. How do you determine the uncertainty in a calculated result when the numbers that go into the calculation are uncertain? The answers to this question is what this subject of error propagation is all about, and to learn more, you could consult the following links. (Of course, when you become sufficiently advanced, even these methods will be insufficient, and then you will need to learn more about probability and statistics.)

Coping with uncertain measurements is extremely important in science, because observation and experiment are the ultimate authorities in science. Testing scientific theories, and evaluating their experimental support, requires a good understanding of how reliable measurements are.

significant figures

propagation of uncertainty

• SI units

Having a consistent system that is easy to work with is essential for the development of science. Without a consistent system of units, how would measurements made in one part of the world be compared with and understood by researchers in another part of the world? Without standardized units, that are the same everywhere and always, how can we be sure that measurements made today are consistent with those made last week?

The story of units used in science, and how their definitions and related conventions came to be and have changed over the years, is an interesting one, and well worth studying by science students. For more information, check out the following link:

SI units

• vectors, scalars, and beyond

What exactly is a vector? Yes, I know, we all learn that a vector is an arrow in high school, but as your understanding grows, you will encounter more advanced perspectives on vectors. It's worth thinking about this question now.

It's also worth noting that a vector is a mathematical concept. In mathematics, a vector is defined as a member of a collection of things that satisfy certain properties. Fine, but how do we know that physical quantities, such as positions, velocities, forces, and so on, can be modelled by vectors? How do we know that the physical quantities really correspond to the mathematical quantities that we use to model them? This is all worth pondering over now.

Vectors in mathematics and physics

As you advance to higher levels of physics, you'll learn that there are many more geometric concepts that are very useful in physics. For example, when you apply a force to a small cube, the response to the force may be different from the different faces of the cube. To describe this, we need a mathematical gadget that has three different components for each of the three different spatial directions. This mathematical concept is a type of tensor, which is a generalization of a vector. You'll eventually encounter tensors if you study continuum mechanics, fluid mechanics, general relativity, or particle physics. If you would like to learn more, check out the following links:

intro to tensors (NASA)

tensors (wikipedia)

tensors (3M)

• the scientific method

(I'll include some comments about science and the scientific method here soon. Check back in a day or two.)