Level 1 — RecognitionOscillations & Waves

Oscillations & Waves

⏱ 20 minutes30 marksprintable — key stays hidden on paper

Level 1: Recognition Test

Time limit: 20 minutes Total marks: 30 Instructions: Answer all questions. For True/False, a correct justification is required for full marks.

Section A — Multiple Choice (1 mark each) [10 marks]

Q1. In simple harmonic motion, the restoring force is: (a) constant in magnitude (b) proportional to displacement and directed towards equilibrium (c) proportional to velocity (d) proportional to acceleration squared

Q2. The angular frequency $\omega$ , period $T$ and frequency $f$ are related by: (a) $\omega = 2\pi f = \dfrac{2\pi}{T}$ (b) $\omega = \dfrac{f}{2\pi}$ (c) $\omega = \dfrac{T}{2\pi}$ (d) $\omega = 2\pi T$

Q3. For a mass on a spring in SHM, the velocity is maximum when: (a) $x = A$ (b) $x = A/2$ (c) $x = 0$ (d) $x = -A$

Q4. The period of a simple pendulum of length $L$ is: (a) $2\pi\sqrt{g/L}$ (b) $2\pi\sqrt{L/g}$ (c) $\dfrac{1}{2\pi}\sqrt{L/g}$ (d) $2\pi L g$

Q5. In a longitudinal wave, the particle oscillations are: (a) perpendicular to wave propagation (b) parallel to wave propagation (c) circular (d) zero

Q6. The condition for constructive interference (path difference $\Delta$ ) is: (a) $\Delta = (n+\tfrac{1}{2})\lambda$ (b) $\Delta = n\lambda$ (c) $\Delta = \lambda/4$ (d) $\Delta = n\lambda/3$

Q7. A critically damped oscillator: (a) oscillates forever (b) returns to equilibrium in the shortest time without oscillating (c) overshoots many times (d) has no damping

Q8. The beat frequency produced by two waves of frequencies $f_1$ and $f_2$ is: (a) $f_1 + f_2$ (b) $\sqrt{f_1 f_2}$ (c) $|f_1 - f_2|$ (d) $\tfrac{1}{2}(f_1+f_2)$

Q9. The Mach number is defined as: (a) $M = c/v$ (sound speed / source speed) (b) $M = v/c$ (source speed / sound speed) (c) $M = v \cdot c$ (d) $M = v - c$

Q10. A quantity that stays constant throughout the motion of an ideal SHM oscillator is: (a) kinetic energy (b) potential energy (c) total mechanical energy (d) displacement

Section B — Matching (1 mark each) [8 marks]

Q11. Match each term with its correct expression/description:

#	Term		Expression / Description
i	SHM displacement	P	$\tfrac{1}{2}kA^2$
ii	Total energy in SHM	Q	$x = A\cos(\omega t + \phi)$
iii	Wave speed	R	point of zero displacement in a standing wave
iv	Node	S	$v = f\lambda$

Q12. Match the oscillator/wave concept with its keyword:

#	Concept		Keyword
i	Resonance	P	energy loss, amplitude decays
ii	Damping	Q	driving frequency = natural frequency
iii	Overtone	R	frequency above the fundamental
iv	Q factor	S	sharpness of resonance / quality

Section C — True / False with Justification (2 marks each) [12 marks]

(1 mark correct T/F, 1 mark justification)

Q13. In SHM, acceleration is maximum at the equilibrium position.

Q14. For a transverse wave on a string, wave speed depends on the tension and the linear mass density.

Q15. At resonance in a lightly damped forced oscillator, the amplitude is at its maximum.

Q16. The frequency of a wave changes when it passes from one medium to another.

Q17. When a source of sound moves towards a stationary observer, the observed frequency is lower than the emitted frequency.

Q18. Doubling the amplitude of an SHM oscillator doubles its total energy.

Answer keyMark scheme & solutions

Section A — MCQ (1 mark each)

Q1 — (b). SHM is defined by $F=-kx$ : force proportional to displacement, opposite in direction. (1)

Q2 — (a). $\omega=2\pi f$ and $f=1/T$ , so $\omega=2\pi/T$ . (1)

Q3 — (c). From $v=\omega\sqrt{A^2-x^2}$ , $v$ is maximum ( $=\omega A$ ) at $x=0$ . (1)

Q4 — (b). Small-angle pendulum: $T=2\pi\sqrt{L/g}$ . (1)

Q5 — (b). Longitudinal waves oscillate parallel to propagation (e.g. sound). (1)

Q6 — (b). Constructive interference: whole-number of wavelengths, $\Delta=n\lambda$ . (1)

Q7 — (b). Critical damping returns to equilibrium fastest without oscillating. (1)

Q8 — (c). Beat frequency $=|f_1-f_2|$ . (1)

Q9 — (b). $M=v/c$ = source speed ÷ sound speed. (1)

Q10 — (c). KE and PE interchange, but total $E=\tfrac{1}{2}kA^2$ is constant. (1)

Section B — Matching (1 mark each pairing)

Q11: i→Q, ii→P, iii→S, iv→R (1 each = 4)

Q12: i→Q, ii→P, iii→R, iv→S (1 each = 4)

Section C — True/False with Justification

Q13 — FALSE. (1) Acceleration $a=-\omega^2 x$ is proportional to $x$ ; it is zero at equilibrium and maximum at $x=\pm A$ . (1)

Q14 — TRUE. (1) Wave speed on a string $v=\sqrt{T/\mu}$ , depending on tension $T$ and linear density $\mu$ . (1)

Q15 — TRUE. (1) Forced oscillation amplitude peaks when the driving frequency approaches the natural frequency (resonance), especially sharp for low damping (high Q). (1)

Q16 — FALSE. (1) Frequency is set by the source and is unchanged; wave speed and wavelength change between media ( $v=f\lambda$ , $f$ constant). (1)

Q17 — FALSE. (1) Source approaching → wavefronts compressed → observed frequency higher ( $f'=f\,c/(c-v_s)$ ). (1)

Q18 — FALSE. (1) $E=\tfrac{1}{2}kA^2\propto A^2$ , so doubling $A$ quadruples the energy. (1)

[
  {"claim":"v_max occurs at x=0 for v=omega*sqrt(A^2-x^2)",
   "code":"A,w,x=symbols('A w x',positive=True); v=w*sqrt(A**2-x**2); result = (v.subs(x,0)==w*A) and (v.subs(x,A)==0)"},
  {"claim":"Total SHM energy is constant = kA^2/2",
   "code":"k,A,w,t,phi=symbols('k A w t phi',positive=True); m=k/w**2; x=A*cos(w*t+phi); v=diff(x,t); E=simplify(Rational(1,2)*m*v**2+Rational(1,2)*k*x**2); result = simplify(E-Rational(1,2)*k*A**2)==0"},
  {"claim":"Doubling amplitude quadruples energy",
   "code":"k,A=symbols('k A',positive=True); E1=Rational(1,2)*k*A**2; E2=Rational(1,2)*k*(2*A)**2; result = simplify(E2/E1)==4"},
  {"claim":"Approaching source raises observed frequency",
   "code":"f,c,vs=symbols('f c vs',positive=True); fp=f*c/(c-vs); result = simplify(fp - f) .subs({f:1,c:340,vs:30}) > 0"}
]