Course syllabus  

We reviewed some properties of springs from the first semester of the course. For instance, the formulas for the restoration force (F = - k x) and the spring potential energy (k x² / 2), where k is the spring constant, which has units of Newtons per meter.

We proceeded to consider the position of the mass on a spring as a function of time: x(t). We looked at the case of a mass attached to a horizontal spring on a frictionless surface, which is stretched and released. we sketched out the expected behavior and noted its resemblance to the function cosine.

Started with a guess of cosine, we searched for a function that satisfies

F = m a

- k x = m d² x / d t²

Eventually we came up with the function: x(t) = x_m cos( w t + f₀ ), where x_m is the amplitude (units: meters), w is the angular velocity/frequency (units: radians per second), and f₀ is the phase constant (units: radians).

We found that the maximum velocity is x_mw and the maximum acceleration is x_mw².

Finally we tested that the function x(t) = x_m cos( w t + f₀ ) satisfies conservation of energy.

We reviewed the material covered on Tuesday. Then we considered the fact that we had two expressions for angular frequency w (namely, w = 2p/T and w = (k/m)^1/2).

I don't think we made the distinction between the "frequency" (f=1/T) and the "angular frequency" (w = 2p/T). The former has units of "cycles per second," while the latter has units of "radians per second." Both "cycles" and "radians" inform us how we are thinking about repetitive (periodic) behavior, but they are what I call "fake units." They are not "physical" units like "meters" or "seconds" but more abstract or geometric units. So the "physical" units of both frequency and angular frequency are inverse seconds, a.k.a. Hertz.

We solved some mass-on-a-spring problems.

We started our discussion of the pendulum. Its equilibrium was when the mass hung vertically downward. The displacement from equilibrium is measured by having an angle between the pendulum and the vertical. The mass then has two forces: its weight acting vertically downward and the tension acting along the string, which is radially. We decided to use a radial-tangential coordinate system (as opposed to a horizontal-vertical coordinate system) because the motion is along the circle, i.e. tangential. Hence we proceeded to break the weight into radial and tangential components, and we started to consider an expression for the tangential acceleration to use in F_tan = m a_tan. But we did not finish.