Question

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 (ω is any positive constant) a. sign ωt - cos ωt b. sin² ωt c. 3 Cos (π/4 - ωt) d. cos ωt + cos 3ωt + cos 5ωt e. exp(-ω²t²) f. 1 + ωt + ω²t²

Answer 1

(a) $sin\omega t-cos\omega t$
= $\sqrt{2}\left(sin\omega t\frac{1}{\sqrt{2}} - cos\omega t\frac{1}{\sqrt{2}}\right)$
= $\sqrt{2}(sin\omega tcos\pi/4-cos\omega tsin\pi/4)$
= $\sqrt{2}sin(\omega t-pi/4)$
hence, it is simple harmonic motion. and its period = 2π/$\omega$

(b) sin³ωt = 1/3(3sinωt - sin3ωt) [ from trigonometric formula ]
each term here, sinωt and sin3ωt represent SHM. But sin³ωt is the result of superposition of two SHMs. Hence, it is only periodic not SHM. Its time period is 2π/ω.

(c) It can be seen that it represents an SHM with a time period of 2π/ω.

(d) It represents periodic motion but not SHM. Its time period is 2π/ω.

(e) An exponential function never repeats itself. Hence, it is a non-periodic motion.

(f) It clearly represents a non-periodic motion.

Answer 2

Answer:

$(a) sin\omega t-cos\omega tsinωt−cosωt </p><p>= \sqrt{2}\left(sin\omega t\frac{1}{\sqrt{2}} - cos\omega t\frac{1}{\sqrt{2}}\right)2(sinωt21−cosωt21) </p><p>= \sqrt{2}(sin\omega tcos\pi/4-cos\omega tsin\pi/4)2(sinωtcosπ/4−cosωtsinπ/4) </p><p>= \sqrt{2}sin(\omega t-pi/4)2sin(ωt−pi/4) </p><p>hence, it is simple harmonic motion. and its period = 2π/\omegaω </p><p></p><p>(b) sin³ωt = 1/3(3sinωt - sin3ωt) [ from trigonometric formula ] </p><p>each term here, sinωt and sin3ωt represent SHM. But sin³ωt is the result of superposition of two SHMs. Hence, it is only periodic not SHM. Its time period is 2π/ω.</p><p></p><p>(c) It can be seen that it represents an SHM with a time period of 2π/ω.</p><p></p><p>(d) It represents periodic motion but not SHM. Its time period is 2π/ω.</p><p></p><p>(e) An exponential function never repeats itself. Hence, it is a non-periodic motion.</p><p></p><p>(f) It clearly represents a non-periodic motion$