(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.
Explanation
(a)(i) K is measured in m\(^{-1}\) and w rad s\(^{-1}\) (ii) \(\frac{w}{k}\) represents speed of wave w = 2\(\pi\)f; and k = \(\frac{2 \pi}{\pi}\) \(\frac{w}{k} \frac{2 \pi f}{2 \pi/\lambda} = f\lambda\) But V = f\(\lambda\) (iii) The direction of travel of the wave :s in the positive x - direction.
(b)(i) Beats are periodic rise and fall in amplitude of the sound produced when two notes of nearly equal frequencies are sounded together. (ii) Beat frequency = \(f_1 - f_2\) = 259 - 256 = 3Hz (iii) f = \(\frac{1}{2L} \sqrt{T}{m}\) 259 = \(\frac{1}{2L} \sqrt{\frac{1200}{0.03}}\) L = 0.386m
(c) 1. Both have converging or convex lenses. 2. Their interiors are black. 3. Both form real, inverted, and diminished image. 4. Images of objects are formed on the retina of the eye and the film of the camera.
