(a) On which day would sound wave travel faster: on a hot or cold day? Explain.
(c) A ray of light is incident on a surface of a ectangular glass prism of refractive index 1.5 illustrated in the diagram below.
(i) Copy the diagram a label the angles of: (\(\alpha\)) Incidence (x); (\(\beta\)) Reflection (y); (\(\gamma\)) refraction (z); with t glass letters indicated.
(ii) Calculate the angle refraction to the nearest whole number.
(d) A sonomesr wire vibrates in simple harmoi motion with a maximum amplitude of 1.0 cm. Calculate the frequency of vibration of the wire, giv that the magnitade of the maximum acceleration of the wire is 980ms\(^{-2}\). [\(\pi \frac{22}{7}\)]
(a) Faster on a hot or cold day?
Sound travels faster on a hot day. The speed of sound in air increases with temperature because the air molecules move faster and transmit the disturbance more rapidly (the speed is proportional to \(\sqrt{T}\), where \(T\) is the absolute temperature).
(b) Why megaphones are shaped like funnels
The funnel (conical) shape channels the sound energy and directs it forward in a narrow beam instead of allowing it to spread out in all directions. This concentrates the sound waves, so the sound is louder and carries farther in the intended direction.
(c) Refraction at the air-glass surface (refractive index \(n = 1.5\), angle of incidence read from the diagram \(= 30^{\circ}\))
(i) In the copied diagram: the angle of incidence (x) is between the incident ray and the normal in the air; the angle of reflection (y) is between the reflected ray and the normal (in air, equal to x); the angle of refraction (z) is between the refracted ray and the normal inside the glass.
(ii) Applying Snell's law from air into glass:
\[ n = \frac{\sin i}{\sin r} \;\Rightarrow\; \sin r = \frac{\sin i}{n} = \frac{\sin 30^{\circ}}{1.5} = \frac{0.5}{1.5} = 0.3333 \]
\[ r = \sin^{-1}(0.3333) = 19.47^{\circ} \approx 19^{\circ} \]
(d) Frequency of the vibrating sonometer wire (SHM)
Amplitude \(A = 1.0\ \text{cm} = 0.01\ \text{m}\); maximum acceleration \(a_{max} = 980\ \text{m s}^{-2}\). For SHM, \(a_{max} = \omega^{2} A\):
\[ \omega^{2} = \frac{a_{max}}{A} = \frac{980}{0.01} = 98000 \]
\[ \omega = \sqrt{98000} = 313.05\ \text{rad s}^{-1} \]
Since \(\omega = 2\pi f\), with \(\pi = \dfrac{22}{7}\):
\[ f = \frac{\omega}{2\pi} = \frac{313.05}{2 \times \tfrac{22}{7}} = \frac{313.05}{6.286} \approx 49.8\ \text{Hz} \approx 50\ \text{Hz} \]
