K A Tsokos Solutions for Chapter: Waves, Exercise 5: Test yourself

A pipe with both ends open has two consecutive harmonics of frequency $300 \mathrm{Hz}$ and $360 \mathrm{Hz}$. Determine the length of the pipe. (Take the speed of sound to be $340 \mathrm{m} {\mathrm{s}}^{-1}$.

A pipe $X$ with both ends open and a pipe $Y$with one open and one closed end have the same frequency in the first harmonic. Calculate the ratio of the length of the pipe $X$ to that of pipe$Y$.

If you walk at one step a second holding a cup of water (diameter $8 \mathrm{cm}$) the water will spill out of the cup. Use this information to estimate the speed of the waves in water.

Consider a string with both ends fixed. A standing wave in the second harmonic mode is established on the string, as shown in the diagram. The speed of wave is $180 \mathrm{m} {\mathrm{s}}^{-1}$.

Consider the vibrations of two points on the string, P and Q. The displacement of point P is given by the equation $y=5.0 \cos \left(45\pi t\right)$ where $y$in $\mathrm{mm}$ and $t$is in seconds. Calculate the length of the string.

Consider a string with both ends fixed. A standing wave in the second harmonic mode is established on the string, as shown in the diagram.

Consider the vibrations of two points on the string, P and Q. The displacement of point P is given by the equation $y=5.0 \cos \left(45\pi t\right)$ where $y$in $\mathrm{mm}$ and $t$is in seconds. State the phase difference between the oscillation of point P and that of point Q. Hence write down the equation giving the displacement of point Q.

A horizontal aluminium rod of length $1.2 \mathrm{m}$ is hit sharply with a hammer. The hammer rebounds from the rod $0.18 ms$ later. Explain why the hammer rebounds.

A horizontal aluminium rod of length $1.2 \mathrm{m}$ is hit sharply with a hammer. The hammer rebounds from the rod $0.18 \mathrm{ms}$ later. Calculate the speed of sound in aluminium.

A horizontal aluminium rod of length $1.2 \mathrm{m}$ is hit sharply with a hammer. The hammer rebounds from the rod $0.18 \mathrm{ms}$ later. Calculate the speed of sound in aluminium. The hammer created a longitudinal standing wave in the rod. Estimate the frequency of the sound wave by assuming that the rod vibrates in the first harmonic.