340 Class Notes, Aug 30

Phy 340 class notes for Fall, 1999

Aug 25	Sep 8	Oct 4	Nov 1	Phy 340 page
Aug 30	Sep 13	Oct 11	Nov 8
	Sep 20	Oct 18	Nov 15
	Sep 27	Oct 25	Nov 22
			Nov 29

Phy 340, Week 2

30 Aug Do more analysis of the terms that appear in the equation for the acceleration in polar coordinates. The third term is easy to interpret (now it's easy to interpret) as the same acceleration you would get in straight-line motion, d²s/ dt², where s is the distance measured along the arc of the circle.

Coordinate systems used in three dimensions. Rectangular, cylindrical, and spherical. Next, the corresponding basis vectors that are used to describe motion in these systems.

Newton's equations of motion. What is mass? Define it in more than one way, and see what is needed for the definitions to make sense. Note the the standard definition is only approximately a constant, the more correct definition is more complex, but yields a simpler resulting mass.

Newton's first law as a definition of an inertial frame. The second law has the assumption of such an inertial frame as a hypothesis before you can conclude that F = dp/ dt.

1 Sep How to check to see if a result makes sense: 1. Dimensions 2. Vectors not equal to scalars 3. Special or limiting cases. Do an example for motion in one dimension. A point mass is subject to a force F₀ cos( w t ), with the initial conditions that at time zero, the coordinate x=0 and the velocity v_x is zero. Before solving the whole problem, notice one special case. If the frequency, w = 0, then this is a constant force, giving a constant acceleration. This is a familiar elementary problem, so you can write the answer to that almost immediately. v_x = ( F₀ / m ) t, and x = ( F₀ / m ) t²/2.

Now do the whole problem, putting in the initial conditions to evaluate the constants of integration. Draw some graphs to see what this solution looks like; check its dimensions; and finally see what happens when the frequency vanishes. This leads to indeterminate forms, 0/ 0 or infinity minus infinity.

Before proceeding, there are some tools that must be in everyone's utility belt. There are certain infinite series, power series, that you will learn to know and love.
sine, cosine, exponential, binomial expansion, logarithm, hyperbolic sine and hyperbolic cosine.
The last two come up less often, but they're still useful.

Use the series for the sine and cosine to analyze the limiting case of small frequency for the problem I just did. It very quickly reduces to the simple constant acceleration result.

3 Sep A diversion into index mechanics. Show how some common manipulations with vectors can be handled in the notation of indices. Introduce the Kronecker d_ij symbol and the alternating symbol e_ijk. The summation convention dicates that when as index appears twice in a single term, it is to be summed from 1 to 3. Vectors themselves have components with respect to some fixed basis (assumed to be orthonormal here), so you have

A = S A_i e_i where the basis vectors are e₁, e₂, and e₃. This notation for the basis provides a more systematic form than the traditional i's, j's, and k's. The summation convention says that the explicit summation sign is suppressed and understood. When you do something such as a scalar product, it becomes in this notation,

A·B = A_ie_i·B_je_j = A_iB_je_i·e_j = A_iB_jd_ij = A_iB_i

A cross product becomes

( A × B )_i = e_ijk A_j B_k.

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