🌊 بنك الأسئلة: الاهتزازات والأمواج

📌 Hooke's restoring-force law: \(F=-kx\); the negative sign means the restoring force always opposes the displacement direction (pushing the body back toward equilibrium), while its magnitude is directly proportional to the displacement's magnitude.

📌 At the equilibrium position: displacement \(x=0\), so the elastic potential energy \(Pe_s=\tfrac12kx^2=0\), while speed is maximum so kinetic energy \(KE>0\) (all the total energy is kinetic at this instant).

📌 Extension: \(x=L_2-L_1=5\,cm=0.05\,m\)

📌 At equilibrium, the weight equals the spring force: \(k=\dfrac{mg}{x}=\dfrac{0.6\times9.81}{0.05}\approx117.7\,N/m\)

📌 Elastic PE = area under the F-x line (a triangle): from the graph, at max extension \(x\approx0.08\,m\), \(F\approx24\,N\).

📌 Area: \(Pe_s=\tfrac12\times24\times0.08\approx0.96\,J\approx1\,J\)

📌 During one full period: the body travels \(0\to A\to0\to-A\to0\), a total distance \(=4A\).

📌 Solving: \(4A=0.4\Rightarrow A=0.1\,m\)

📌 A longitudinal wave: particle vibration is parallel to the wave's direction of travel (compressions and rarefactions), exactly what happens with sound (vibrating air molecules). Light (transverse EM) and rope/ocean-surface waves (transverse or surface) are not longitudinal.

📌 Simple pendulum period: \(T=2\pi\sqrt{L/g}\Rightarrow g=\dfrac{4\pi^2L}{T^2}=\dfrac{4\pi^2(2)}{3.6^2}\approx6.09\,m/s^2\)

📌 Since \(T\propto1/\sqrt g\): \(\dfrac{T_{earth}}{T_{moon}}=\sqrt{\dfrac{g_{moon}}{g_{earth}}}=\sqrt{\dfrac16}\approx0.408\)

📌 Definition of wavelength: the distance between any two consecutive points vibrating in exactly the same phase (same displacement and direction of motion at every instant). Points \(B\) and \(E\) sit at the same relative phase position in two consecutive cycles, so the distance between them equals one full wavelength.

📌 Wave speed depends on the medium's properties (the rope itself), not on the source: since the rope itself didn't change, \(v=8\,m/s\) stays the same.

📌 New frequency: \(f=\dfrac{v}{\lambda}=\dfrac{8}{1.5}\approx5.33\,Hz\) (changed because the second student's hand shook the rope at a different rate)

📌 From the graph: wavelength \(\lambda\approx10\,cm=0.1\,m\)

📌 Wave equation: \(f=\dfrac{v}{\lambda}=\dfrac{4}{0.1}=40\,Hz\)

📌 From the graph: period \(T\approx20\,s\Rightarrow f=\dfrac1T=0.05\,Hz\)

📌 Wave equation: \(\lambda=\dfrac{v}{f}=\dfrac{3}{0.05}=60\,m\)

📌 Standing-wave condition on a rope of length L (both ends fixed): \(L=\dfrac{n\lambda}{2}\Rightarrow\lambda=\dfrac{2L}{n}=\dfrac{4}{n}\) for integer \(n=1,2,3,...\)

📌 Possible values: \(n=1\Rightarrow4\,m\), \(n=2\Rightarrow2\,m\), \(n=3\Rightarrow1.33\,m\); but \(\lambda=3\,m\) requires \(n=4/3\) (not an integer), so it cannot form a standing wave.

📌 Sound travels the distance there and back: total distance \(=2X=7250\,m\)

📌 Speed: \(v=\dfrac{2X}{t}=\dfrac{7250}{5}=1450\,m/s\)

📌 Reflection doesn't change the medium: the wave stays in the same medium (the same water tank) before and after reflecting, so its speed, wavelength, and frequency all stay the same (only the direction of travel, and possibly the phase, changes).

📌 Frequency always stays constant across refraction: frequency is set only by the source and never changes when a wave crosses into another medium, so \(f_1=f_2\) always — meaning \(f_1>f_2\) is the false statement.

📌 The other statements are actually true: speed decreases in shallow water (\(v_1>v_2\) is true), and since \(\lambda=v/f\) with f constant, wavelength decreases proportionally (\(\lambda_1>\lambda_2\) is true), and the wave bends toward the normal entering the slower medium (\(\theta_1>\theta_2\) is true).

📌 \(T=2\pi\sqrt{L/g}\Rightarrow T\propto\sqrt L\): if the length becomes \(4L\), then \(T_{new}=2\pi\sqrt{4L/g}=2T\)

📌 By definition: frequency is the number of complete cycles per second, the reciprocal of the period: \(f=1/T\), regardless of the type of periodic motion.

📌 Angular frequency: \(\omega=\dfrac{2\pi}{T}=\dfrac{2\pi}{0.5}=4\pi\approx12.57\,rad/s\)

📌 Simple pendulum period formula: \(T=2\pi\sqrt{L/g}\) — notice that mass \(m\) does not appear in the formula at all, so a simple pendulum's period is completely independent of its mass.

📌 Mass-spring system period formula: \(T=2\pi\sqrt{m/k}\Rightarrow T\propto\sqrt m\)

📌 With the mass quadrupled: \(T_{new}=2\pi\sqrt{4m/k}=2T\)

📌 Total energy: \(E=\tfrac12kA^2\), and it stays constant throughout the motion (continuously exchanged between kinetic and elastic potential energy, but their sum is constant).

📌 At the equilibrium position: all the system's total energy is kinetic (elastic potential energy is zero there), so speed is maximum exactly at this point, and becomes zero at maximum displacement.

📌 From \(a=-\omega^2x\): acceleration is directly proportional to displacement but always opposite in direction — i.e., it always points toward the equilibrium position, which is the essence of the restoring force that produces simple harmonic motion.

📌 \(E=\tfrac12kA^2\Rightarrow E\propto A^2\): doubling the amplitude \(A\to2A\) gives \(E_{new}=\tfrac12k(2A)^2=4E\)

📌 Series connection: \(\dfrac1{k_{eq}}=\dfrac1{k_1}+\dfrac1{k_2}\Rightarrow k_{eq}\) is always smaller than the smaller of the two springs — since a series connection makes the system easier to stretch (less stiff) than either spring alone.

📌 Mass-spring period formula: \(T=2\pi\sqrt{\dfrac{m}{k}}=2\pi\sqrt{\dfrac{2}{50}}\approx1.26\,s\)

📌 Wave equation: \(\lambda=\dfrac{v}{f}=\dfrac{340}{170}=2\,m\)

📌 Definition of a transverse wave: particle vibration is perpendicular to the direction energy travels, as in rope waves, light, and electromagnetic waves generally.

📌 Mechanical vs. electromagnetic waves: sound, water waves, and rope waves are all mechanical waves that need a material medium to travel through. Electromagnetic waves (like light), however, arise from oscillating electric and magnetic fields and need no medium at all, so they travel through vacuum (as sunlight reaches Earth through space).

📌 The Doppler effect: when a sound source approaches an observer, the wave crests reaching them get compressed into a shorter time, so the observer perceives a higher frequency than the actual emitted one. The opposite happens as the source moves away.

📌 Constructive interference: when two waves are exactly in phase, their displacements add (superposition principle), producing a larger combined displacement (greater amplitude) than either wave alone.

📌 Destructive interference: when the two waves are completely out of phase with equal amplitudes, they exactly cancel each other at the point of overlap (resultant = zero).

📌 Diffraction: the phenomenon of a wave bending around the edges of barriers or when passing through narrow openings, becoming more pronounced as the opening's width approaches the wavelength itself.

📌 A node: a fixed point in a standing wave produced by continuous total destructive interference there (between the incident and reflected waves), so its displacement stays zero at all times. An antinode is the point of maximum displacement (constructive interference).

📌 Resonance: when a periodic external driving force's frequency matches a vibrating system's natural frequency, energy is pumped into it with maximum efficiency each cycle, building up its vibration amplitude to the largest possible value (example: pushing a swing at just the right timing).

📌 Wave equation: \(f=\dfrac{v}{\lambda}=\dfrac{330}{1.5}=220\,Hz\)

📌 From the graph the amplitude and period: \(A=0.1\,m,\quad T=2\,s\Rightarrow\omega=\dfrac{2\pi}{T}=\pi\,rad/s\)

📌 The maximum speed occurs at the equilibrium position: \(v_{max}=A\omega=0.1\times\pi=0.31\,m/s\)

📌 From the graph: \(A=0.2\,m,\quad T=4\,s\Rightarrow\omega=\dfrac{2\pi}{4}=\dfrac{\pi}{2}\,rad/s\)

📌 The maximum acceleration occurs at maximum displacement: \(a_{max}=A\omega^2=0.2\times\left(\dfrac{\pi}{2}\right)^2=0.49\,m/s^2\)

📌 Conservation of mechanical energy: \(\tfrac12kA^2=\tfrac12mv^2+\tfrac12kx^2\)

📌 Solve for \(v\): \(v=\sqrt{\dfrac{k(A^2-x^2)}{m}}=\sqrt{\dfrac{200(0.01-0.0025)}{0.5}}=\sqrt{3}=1.73\,m/s\)

📌 At maximum displacement all the energy is potential: \(E=\tfrac12kA^2\)

📌 Substituting: \(k=\dfrac{2E}{A^2}=\dfrac{2(2)}{0.2^2}=100\,N/m\)

📌 The series spring constant: \(\dfrac1{k}=\dfrac1{300}+\dfrac1{600}=\dfrac{3}{600}\Rightarrow k=200\,N/m\)

📌 The period: \(T=2\pi\sqrt{\dfrac mk}=2\pi\sqrt{\dfrac{2}{200}}=0.63\,s\)

📌 From \(T=2\pi\sqrt{L/g}\): \(L=\dfrac{gT^2}{4\pi^2}\)

📌 Substituting: \(L=\dfrac{9.8\times4}{39.5}=0.99\,m\)

📌 Effective gravity in an upward-accelerating elevator: \(g_{eff}=g+a=2g\)

📌 The period: \(T'=2\pi\sqrt{\dfrac{L}{2g}}=\dfrac{T}{\sqrt2}\)

📌 The wavelength is the distance between two successive crests: \(\lambda=0.6\,m\)

📌 Wave speed: \(v=f\lambda=5\times0.6=3.0\,m/s\)

📌 The period from the graph (one full oscillation): \(T=40\,ms=0.04\,s\Rightarrow f=\dfrac1T=25\,Hz\)

📌 The wavelength: \(\lambda=\dfrac vf=\dfrac{20}{25}=0.8\,m\)

📌 The figure shows three loops: the length equals three half-wavelengths: \(L=\dfrac{3\lambda}{2}\Rightarrow\lambda=\dfrac{2L}{3}=0.8\,m\)

📌 The frequency: \(f=\dfrac v\lambda=\dfrac{60}{0.8}=75\,Hz\)

📌 The tension = the weight of the hanging mass, and the mass per unit length: \(T=mg=100\,N,\quad\mu=\dfrac{0.02}{2}=0.01\,kg/m\)

📌 The speed: \(v=\sqrt{\dfrac{T}{\mu}}=\sqrt{\dfrac{100}{0.01}}=100\,m/s\)

📌 The frequency does not change across the boundary (set by the source): \(f=\dfrac{v_1}{\lambda_1}=\dfrac{6}{0.03}=200\,Hz\)

📌 The new wavelength: \(\lambda_2=\dfrac{v_2}{f}=\dfrac{4}{200}=0.02\,m=2\,cm\)

📌 The path difference: \(\Delta=7.5-6.0=1.5\,m=1.5\lambda\)

📌 The path difference is an odd number of half-wavelengths (\(3\times\tfrac\lambda2\)) so crest meets trough: destructive interference.

📌 In a transverse wave the particles oscillate perpendicular to the propagation direction, so they do not move horizontally.

📌 The wave moves right, so P takes the displacement of the point to its left (which is lower) because it sits on an ascending slope toward the crest: P moves down.

📌 In a closed pipe the length equals a quarter of the fundamental wavelength: \(L=\dfrac\lambda4\Rightarrow\lambda=4L=3.4\,m\)

📌 The frequency: \(f_1=\dfrac v\lambda=\dfrac{340}{3.4}=100\,Hz\)

📌 Extension: \(x=L_2-L_1=0.05\,m\)

📌 At equilibrium: \(k=\dfrac{mg}{x}=\dfrac{0.6\times9.81}{0.05}\approx117.7\,N/m\)

📌 From the graph: at max extension \(x\approx0.08\,m\), \(F\approx24\,N\)

📌 Elastic PE = triangle area under the line: \(Pe_s\approx0.96\,J\approx1\,J\)

📌 (a) Force at max displacement (Hooke's law): \(F=kX=200\times0.1=20\,N\)

📌 (b) Total system energy (conserved): \(E=\tfrac12kX^2=\tfrac12(200)(0.1)^2=1\,J\)

📌 At equilibrium: elastic PE is zero, so all energy is kinetic: \(KE=E=1\,J\)

📌 (a) From \(T=2\pi\sqrt{L/g}\): \(g=\dfrac{4\pi^2L}{T^2}\approx6.1\,m/s^2\)

📌 (b) Frequency: \(f=\dfrac1T\approx0.28\,Hz\)

📌 (a) Amplitude (read directly off the displacement axis): \(A=5\,m\)

📌 (b) Wavelength (from the pattern's repeat on the distance axis): \(\lambda\approx10\,m\)

📌 Frequency: \(f=\dfrac{v}{\lambda}=\dfrac{4}{10}=0.4\,Hz\)