Oscillations

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5 of 194

Question 5 of 194

Match List-I with List-II.
List-IList-IIA. sin⁡2ωt\sin^2 \omega tI. Periodic with time period T=πωT = \dfrac{\pi}{\omega} but not simple harmonic motion (SHM)B. sin⁡3(2ωt)\sin^3(2\omega t)II. Periodic with time period T=2πωT = \dfrac{2\pi}{\omega} but Not SHMC. sin⁡(ωt)+cos⁡(πωt)\sin(\omega t) + \cos(\pi \omega t)III. Periodic with time period T=πωT = \dfrac{\pi}{\omega} and SHMD. cos⁡ωt+cos⁡2ωt\cos\omega t + \cos 2\omega tIV. Non-periodic
Choose the correct answer from the options given below :

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Answer: (A) A-III, B-I, C-IV, D-II

For A: y=sin⁡2ωt=1−cos⁡2ωt2=12−12cos⁡2ωty = \sin^2 \omega t = \dfrac{1 - \cos 2\omega t}{2} = \dfrac{1}{2} - \dfrac{1}{2} \cos 2\omega t

This represents simple harmonic motion (SHM) about the mean position y=12y = \dfrac{1}{2} with an angular frequency of 2ω2\omega. The time period is T=2π2ω=πωT = \dfrac{2\pi}{2\omega} = \dfrac{\pi}{\omega}. Thus, A matches III.

For B: y=sin⁡3(2ωt)=3sin⁡2ωt−sin⁡6ωt4y = \sin^3(2\omega t) = \dfrac{3\sin 2\omega t - \sin 6\omega t}{4}

This is a superposition of two SHMs with angular frequencies 2ω2\omega and 6ω6\omega. It is periodic but not SHM. The time period is the LCM of T1=2π2ω=πωT_1 = \dfrac{2\pi}{2\omega} = \dfrac{\pi}{\omega} and T2=2π6ω=π3ωT_2 = \dfrac{2\pi}{6\omega} = \dfrac{\pi}{3\omega}, which is T=πωT = \dfrac{\pi}{\omega}. Thus, B matches I.

For C: y=sin⁡(ωt)+cos⁡(πωt)y = \sin(\omega t) + \cos(\pi \omega t)

The angular frequencies are ω1=ω\omega_1 = \omega and ω2=πω\omega_2 = \pi \omega. The ratio ω1ω2=1π\dfrac{\omega_1}{\omega_2} = \dfrac{1}{\pi} is an irrational number. Therefore, the function is non-periodic. Thus, C matches IV.

For D: y=cos⁡ωt+cos⁡2ωty = \cos \omega t + \cos 2\omega t

This is a superposition of two SHMs with angular frequencies ω\omega and 2ω2\omega. It is periodic but not SHM. The time period is the LCM of T1=2πωT_1 = \dfrac{2\pi}{\omega} and T2=2π2ω=πωT_2 = \dfrac{2\pi}{2\omega} = \dfrac{\pi}{\omega}, which is T=2πωT = \dfrac{2\pi}{\omega}. Thus, D matches II.

The correct matching is A-III, B-I, C-IV, D-II.

Answer: A-III, B-I, C-IV, D-II

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