A string on a musical instrument is held under tension $T$ and extends from the point $x=0$ to the point $x=L$. The string is overwound with wire in such a way that its mass per unit length $\mu(x)$ increases uniformly from $\mu_0$ at $x=0$ to $\mu_L$ at $x=L$. (a) Find an expression for $\mu(x)$ as a function of $x$ over the range $0 \leq x \leq L$.

A light, cubical container of volume $a^{3}$ is initially filled with a liquid of mass density $\rho$. The cube is initially supported by a light string to form a simple pendulum of length $L_{i}$ measured from the center of mass of the filled container, where $L_{i}>>a .$ The liquid is allowed to flow from the bottom of the container at a constant rate (dM/dt) At any time t, the level of the liquid in the container is h and the length of the pendulum is L (measured relative to the instantaneous center of mass). (a) Find the period of the pendulum as a function of time. (b) What is the period of the pendulum after the liquid completely runs out of the container?

Write a short story about a student your age who has to make an economic choice. In your story, reveal how plentiful wants conflict with scarce resources.

For a certain transverse wave, the distance between two successive crests is 1.20 m, and eight crests pass a given point along the direction of travel every 12.0 s. Calculate the wave speed.

An earthquake on the ocean floor in the Gulf of Alaska produces a tsunami (sometimes incorrectly called a "tidal wave") that reaches Hilo, Hawaii, $4450 \mathrm{~km}$ away, in a time interval of $9 \mathrm{~h} 30 \mathrm{~min}$. Tsunamis have enormous wavelengths (100 to $200 \mathrm{~km}$ ), and the propagation speed for these waves is $v \approx \sqrt{g} d$, where $\bar{d}$ is the average depth of the water. From the information given, find the average wave speed and the average ocean depth between Alaska and Hawaii. (This method was used in 1856 to estimate the average depth of the Pacific Ocean long before soundings were made to give a direct determination.)

If a loop of chain is spun at high speed, it can roll along the ground like a circular hoop without collapsing. Consider a chain of uniform linear mass density $\mu$ whose center of mass travels to the right at a high speed $v_{0}.$ (a) Determine the tension in the chain in terms of $\mu$ and $v_{0}.$ Assume the weight of an individual link is negligible compared to the tension. (b) If the loop rolls over a small bump, the resulting deformation of the chain causes two transverse pulses to propagate along the chain, one moving clockwise and one moving counterclockwise. What is the speed of the pulses traveling along the chain? (c) Through what angle does each pulse travel during the time interval over which the loop makes one revolution?

A bat, moving at 5.00 m/s, is chasing a flying insect. If the bat emits a 40.0-kHz chirp and receives back an echo at 40.4 kHz, (a) what is the speed of the insect? (b) Will the bat be able to catch the insect? Explain.

What happens to the wavelength of a wave on a string when the frequency is doubled? Assume that the tension in the string remains constant.

Does the vertical speed of a segment of a horizontal taut string, through which a wave is traveling, depend on the wave speed?

Why is the following situation impossible? Tsunamis are ocean surface waves that have enormous wavelengths (100 to 200 km), and the propagation speed for these waves is $v \approx \sqrt{g d_{\text {avg }}},$ where $d_{\text {avg }}$ is the average depth of the water. An earthquake on the ocean floor in the Gulf of Alaska produces a tsunami that reaches Hilo, Hawaii, 4450 km away, in a time interval of 5.88 h. (This method was used in 1856 to estimate the average depth of the Pacific Ocean long before soundings were made to give a direct determination.)

It is stated in the earlier problem that a pulse travels from the bottom to the top of a hanging rope of length $L$ in a time interval $\Delta t=2 \sqrt{L / g}$. Use this result to answer the following questions. (It is not necessary to set up any new integrations.) (a) How long does it take for a pulse to travel halfway up the rope? Give your answer as a fraction of the quantity $2 \sqrt{L / g}$. (b) A pulse starts traveling up the rope. How far has it traveled after a time interval $\sqrt{L / g}$?

The given wave function describes A pulse traveling along a string of linear mass density $\mu$.

$$y=\left[A_0 e^{-b x}\right] \sin (k x-\omega t)$$

where the factor in brackets is said to be the amplitude. Calculate the ratio $\mathscr{P}(x) / \mathscr{P}(0)$.

A string of length $L$ consists of two sections. The left half has mass per unit length $\mu=\mu_0 / 2$, while the right has a mass per unit length $\mu^{\prime}=3 \mu=3 \mu_0 / 2$. Tension in the string is $T_0$ - Notice from the data given that this string has the same total mass as a uniform string of length $L$ and mass per unit length $\mu_0$. (a) Find the speeds $\nu$ and $v^{\prime}$ at which transverse pulses travel in the two sections. Express the speeds in terms of $T_0$ and $\mu_0$, and also as multiples of the speed $w_0=\left(T_0 / \mu_0\right)^{1 / 2}$. (b) Find the time interval required for a pulse to travel from one end of the string to the other. Give your result as a multiple of $\Delta t_0=L / v_0-$

A flowerpot is knocked off a window ledge from a height d = 20.0 m above the sidewalk. It falls toward an unsuspecting man of height h = 1.75 m who is standing below. Assume the man requires a time interval of $\Delta t=0.300 s$ to respond to the warning. How close to the sidewalk can the flowerpot fall before it is too late for a warning shouted from the balcony to reach the man in time?