difference between dimensions and units. Dimensions refer to physical characteristics such as mass, length, or time. Units refer to the system of measurement we use to substitute numbers into an equation. Examples are kilograms for mass, feet for length, and minutes for time.4 4 To see that this is the case, first divide both sides of Equation 1.30 by , then let go to zero and note that the left‐hand sides become , , and respectively. On the right‐hand side, becomes for all i. Finally, take the partial derivatives with respect to and the expressions in Equation 1.38 result.
5 5 The definition of the often‐quoted simple pendulum is that it has a massless rigid rod supporting a point mass. The rod is free to swing in a plane about the frictionless point where it is connected to the ground. The only external force is that due to gravity.
6 6 All of these equilibrium states have very interesting stability characteristics but considering stability is outside the scope of what we are doing here. It is sufficient for our purposes to say, without proof, that the equilibrium state between and is stable.
7 7 While the “sum of angles formulae” may be “well known”, they are not easily remembered. However, they can be quickly derived using Euler's equationWe write the equation twice, once for the angle and again for the angle , givingWe then multiply these two equations together giving, on the left‐hand side,and, on the right‐hand side,Equating the real and imaginary parts of these two results gives the “sum of angles” formulae we couldn't remember
8 8 In the case of an angle of ten degrees, for example, we convert the angle to 0.174533 radians. The percentage errors using the linear approximations are only 0.5% on the sine and 1.5% on the cosine. Ten degrees of rotation is a very large angle in the world of vibrations. We are typically looking at fractions of a degree where the linear approximations are very accurate.
9 9 For those readers who already know about the standard form of these equations and are worried about the consequences of the term in square brackets being negative, I suggest you take time to consider what value must have for this term to be negative.
10 10 All of the analysis here has assumed that the motion of the system can be described with a single degree of freedom and therefore a single equation of motion. We will soon see that this is not the case but that the material and methods presented here still apply.
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