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Q13.4: NCERT Physics - Part 2 OSCILLATIONS 13.4 Solution: Which of the following functions of time represent (a) simple…

Which of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic, and (c) non-periodic motion? Give period for each case of periodic motion (_ is any positive constant): (a) sin _t � cos _t (b) sin_ _t (c) 3 cos (�/4 � 2_t) (d) cos _t + cos 3_t + cos 5_t (e) exp (�__t_) (f) 1 + _t + __t_

Step-by-Step Verified Solution

**Answer:** (a) SHM, period 2�/_. (b) Periodic but not SHM, period 2�/_. (c) SHM, period �/_. (d) Periodic but not SHM, period 2�/_. (e) Non-periodic. (f) Non-periodic. **Step-by-Step Explanation:** (a) sin⁡ωt−cos⁡ωt=2sin⁡(ωt−π/4)\sin \omega t - \cos \omega t = \sqrt{2}\sin(\omega t - \pi/4). This is SHM with period 2π/ω2\pi/\omega [citation:2]. (b) sin⁡3ωt=14[3sin⁡ωt−sin⁡3ωt]\sin^3 \omega t = \frac{1}{4}[3\sin \omega t - \sin 3\omega t]. Each term is SHM, but their superposition is periodic, not SHM. Period = 2π/ω2\pi/\omega [citation:2]. (c) 3cos⁡(π/4−2ωt)=3cos⁡(2ωt−π/4)3\cos(\pi/4 - 2\omega t) = 3\cos(2\omega t - \pi/4). This is SHM with period 2π/(2ω)=π/ω2\pi/(2\omega) = \pi/\omega [citation:2]. (d) cos⁡ωt+cos⁡3ωt+cos⁡5ωt\cos \omega t + \cos 3\omega t + \cos 5\omega t. Each term is SHM. Their superposition is periodic, not SHM. Period = 2π/ω2\pi/\omega [citation:2]. (e) exp⁡(−ω2t2)\exp(-\omega^2 t^2) is an exponential function. It does not repeat itself, so it is non-periodic [citation:2]. (f) 1+ωt+ω2t21 + \omega t + \omega^2 t^2 is a polynomial function. It does not repeat itself, so it is non-periodic.

All Questions in this Chapter

Q13.1 Q13.2 Q13.3 Q13.4 Q13.5 Q13.6 Q13.7 Q13.8 Q13.9 Q13.10 Q13.11 Q13.12 Q13.13 Q13.14 Q13.15 Q13.16 Q13.17 Q13.18

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