(a) The equation y = \(\alpha\) sin (wt - kx) represents a plane wave travelling in a medium along the x-direction, being the displacement at the point x at time t.
(i) Given that x is in metres and t is in seconds, state the units of k and w
(ii) What physical quantity does \(\frac{W}{k}\) represent? Justify your answer
(iii) State whether the wave is travelling in the positive or negative x-direction.
(b)(i) What are beats?
(ii) A sonometer wire has a frequency of 259 Hz. It is sounded alongside a tuning fork of frequency 256 Hz. Calculate the beat frequency
(iii) The sonometer wire in (b)(ii) above is under a tension of 1200 N. If a metre of the wire has a mass of 0.03 kg, calculate the length l of the wire when it is vibrating in the fundamental mode.
(c) List two similarities between the human eye and the photographic camera.
(a)(i) Units in \(y = a\sin(\omega t - kx)\): the argument of sine must be a pure number. Since \(t\) is in seconds, \(\omega\) has unit rad s\(^{-1}\) (per second); since \(x\) is in metres, \(k\) has unit rad m\(^{-1}\) (per metre).
(a)(ii) \(\dfrac{\omega}{k}\) represents the speed (velocity) of the wave. Justification: \(\omega = 2\pi f\) and \(k = \dfrac{2\pi}{\lambda}\), so \(\dfrac{\omega}{k} = \dfrac{2\pi f}{2\pi/\lambda} = f\lambda = v\), which is the wave speed.
(a)(iii) The form \((\omega t - kx)\) represents a wave travelling in the positive x-direction.
(b)(i) Beats are the periodic rise and fall in the loudness (amplitude) of sound heard when two notes of slightly different frequencies are sounded together, caused by their alternate reinforcement and cancellation.
(b)(ii) Beat frequency:
\[ f_{\text{beat}} = |259 - 256| = 3\,\text{Hz} \]
(b)(iii) Length of the sonometer wire (fundamental mode): the wave speed on the wire is
\[ v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{1200}{0.03}} = \sqrt{40000} = 200\,\text{ms}^{-1} \]
For the fundamental, \(f = \dfrac{v}{2l}\), so:
\[ l = \frac{v}{2f} = \frac{200}{2\times 259} = 0.386\,\text{m}\ (\approx 38.6\,\text{cm}) \]
(c) Two similarities between the human eye and the photographic camera:
- Both use a converging (convex) lens to form a real, inverted, diminished image of the object.
- Both form the image on a light-sensitive surface (the retina in the eye, the film/sensor in the camera) and both control the amount of light entering through an adjustable aperture (the iris/pupil in the eye, the diaphragm in the camera).
(a)(i) Units in \(y = a\sin(\omega t - kx)\): the argument of sine must be a pure number. Since \(t\) is in seconds, \(\omega\) has unit rad s\(^{-1}\) (per second); since \(x\) is in metres, \(k\) has unit rad m\(^{-1}\) (per metre).
(a)(ii) \(\dfrac{\omega}{k}\) represents the speed (velocity) of the wave. Justification: \(\omega = 2\pi f\) and \(k = \dfrac{2\pi}{\lambda}\), so \(\dfrac{\omega}{k} = \dfrac{2\pi f}{2\pi/\lambda} = f\lambda = v\), which is the wave speed.
(a)(iii) The form \((\omega t - kx)\) represents a wave travelling in the positive x-direction.
(b)(i) Beats are the periodic rise and fall in the loudness (amplitude) of sound heard when two notes of slightly different frequencies are sounded together, caused by their alternate reinforcement and cancellation.
(b)(ii) Beat frequency:
\[ f_{\text{beat}} = |259 - 256| = 3\,\text{Hz} \]
(b)(iii) Length of the sonometer wire (fundamental mode): the wave speed on the wire is
\[ v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{1200}{0.03}} = \sqrt{40000} = 200\,\text{ms}^{-1} \]
For the fundamental, \(f = \dfrac{v}{2l}\), so:
\[ l = \frac{v}{2f} = \frac{200}{2\times 259} = 0.386\,\text{m}\ (\approx 38.6\,\text{cm}) \]
(c) Two similarities between the human eye and the photographic camera:
- Both use a converging (convex) lens to form a real, inverted, diminished image of the object.
- Both form the image on a light-sensitive surface (the retina in the eye, the film/sensor in the camera) and both control the amount of light entering through an adjustable aperture (the iris/pupil in the eye, the diaphragm in the camera).
