Waves and Oscillations Summary Questions

1. We consider a massless spring of strength $k$ and rest length ${\ell }_0$ hanging from a ceiling to which we attach a block of mass $m$ at the bottom end (see below). The whole problem is studied in a terrestrial frame $(O,\hat{j})$ assumed Galilean and friction effects are neglected.
   1. Determine the length of the spring when the block is at mechanical equilibrium.
   2. By taking the derivative of the mechanical energy of the block, find the equation of motion satisfied by the block when it is moving. Do not attempt to solve it.
   3. Show that the expression $y(t)={\ell }_0+\frac{mg}{k}+a\cos ({\omega }_{0}t)$ is a solution to the equation found in part 2 provided that $({\omega }_0{)}^2=\frac{k}{m}$.
   4. What should be the value of $A$ if we want $y(0)={\ell }_0$?

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2. We consider a pendulum made of a mass $m$ rigidly attached to a massless hard wire of length $\ell$ with one end attached to a ceiling as observed from a terrestrial frame of reference assumed Galilean (see below).
   1. Draw a diagram of the forces acting on the ball and give their direction in a frame of your choice.
   2. As the pendulum swings, the ball follows a circular trajectory of radius $\ell$ uniquely characterised by the angle $\theta (t)$. Use this information to give the expression of the mechanical energy of the pendulum as a function of $\theta$ and its time derivative $\dot{\theta }$.
   3. By taking the time derivative of the mechanical energy find the equation of motion for the coordinate $\theta (t)$. Simplify this expression by assuming that $\theta$ is small and thus that $\sin (\theta )\approx \theta$.
   4. Verify that$\theta (t)={\theta }_0\cos ({\omega }_{0}t+\phi )$ is a solution of the equation obtained in part 3, provided that $({\omega }_0{)}^{2=}\frac{g}{\ell }$.

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3. In a Galilean frame $(O,\hat{i},\hat{j})$ on Earth, a car is moving with constant acceleration $a=a_x\hat{i}$. A 300 g bob inside the car is attached to the roof of the car by a wire of negligible mass and is assumed to move with the same acceleration as that of the car. We denote by $\theta$ the angle made by the bob with the vertical direction and by $F$ the force from the wire on the bob.

   1. Show that if $a_x=0$ then according to Newtonian mechanics the bob cannot be hanging only along the vertical direction.
   2. If $\theta =45^{\circ }$, what is the magnitude of the acceleration $a_x$?
   3. If the bob is slightly displaced from its equilibrium position described in (b), explain qualitatively what its motion will be like.

4. A string of length $L$, mass per unit length $\mu$, and tension $T$ is fixed at both ends. The string can support transverse waves, and we consider the situation where standing waves are formed.

   1. Write down the wave equation for a transverse displacement $y(x,t)$ of the string.
   2. Show that the general solution of this wave equation can be written in the form $y(x,t)={\sum }_{n=1}^{\infty }{A}_n\sin \left(\frac{n\pi x}{L}\right)\cos \left({\omega }_{n}t+{\varphi }_n\right)$ where ${\omega }_n$ is the angular frequency of the $n$th harmonic.
   3. Express ${\omega }_n$ in terms of $n$, $L$, $\mu$, and $T$.
   4. If $L=1.5$ m, $\mu =0.02$ kg/m, and $T=10$ N, calculate the frequencies of the first three harmonics.

5. Consider a source of sound moving with velocity $v_s$ in a medium where the speed of sound is $c$. An observer is stationary relative to the medium.

   1. Derive the expression for the observed frequency $f^{\prime }$ when the source is moving directly towards the observer.
   2. What happens to the observed frequency $f^{\prime }$ when the source is moving directly away from the observer?
   3. If $v_s=340$ m/s, $c=343$ m/s, and the source emits a frequency of $f=440$ Hz, what will be the observed frequency when the source is moving towards and away from the observer?