91Ó°ÊÓ

The displacement (from equilibrium) of a particle executing simple harmonic motion mas be either \(y=A \sin \omega t\) or \(y=A \sin (\omega t+\phi)\) depending on our choice of time origin. Show that the average of the kinetic energy of a particle of mass \(m\) (over a period of the motion) is the same for the two formulas (as it must be since both describe the same physical motion). Find the average value of the kinetic energy for the sin \((\omega t+\phi)\) casc in two ways: (a) By selecting the integration limits (as you may by Problem \(+1\) ) so that a change of variable reduces the integral to the \(\sin \omega t\) case. (b) By expanding \(\sin (\omega t+\phi\) ) by the trigonometric addition formulas and using \((5.2)\) to write the average values.

(a) Prove that \(\int_{0}^{\pi / 2} \sin ^{2} x d x=\int_{0}^{x / 2} \cos ^{2} x d x\) by making the change of variable \(x=\frac{1}{2} \pi-t\) in one of the integrals. (b) Use the same method to prove that the averages of \(\sin ^{2}(n \pi x / l)\) and \(\cos ^{2}(n \pi x / l)\) are the same over a period.

A simple pendulum is a small mass \(m\) suspended, as shown, by a (weightless) string. Show that for small oscillations (small \(\theta\) ), both \(\theta\) and \(x\) are sinusoidal functions of time, that is, the motion is simple harmonic. Hint: Write the differential equation \(\mathbf{F}=m \mathbf{a}\) for the particle \(m\). Use the approximation \(\sin \theta=\theta\) for small \(\theta\), and show that \(\theta=A \sin \omega t\) is a solution of your equation. What are \(A\) and \(\omega ?\)

Each of the following functions is given over one period. Sketch several periods of the corresponding periodic function and expand it in an appropriate Fourier series. $$ f(x)= \begin{cases}0, & -\frac{1}{2}