Which of the following examples represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion?a. The rotation of earth about its axisb. Motion of an oscillating mercury column in a U-tubec. Motion of a ball bearing inside a smooth curved bowl when released from a point slightly above the lowermost point.d. General vibrations of a polyatomic molecule about its equilibrium position.
Answers
(a) The rotation of the earth about its axis is a periodic motion. as period of its is 24 hours. But as it is not to and fro motion, it does not represent S.H.M.
(b) The mercury column which oscillates back and forth in a U-tube represents S.H.M.
(c) The motion of the ball bearing inside a smooth curved bowl represents S.H.M as it oscillates to and fro in the bowl.
(d) The general vibrations of polyatomic molecule about its equilibrium position are periodic motion . But they do not represent S.H.M as it is a superposition of SHMs by individual atoms.
Answer:
(a) sin\omega t-cos\omega tsinωt−cosωt
= \sqrt{2}\left(sin\omega t\frac{1}{\sqrt{2}} - cos\omega t\frac{1}{\sqrt{2}}\right)
2
(sinωt
2
1
−cosωt
2
1
)
= \sqrt{2}(sin\omega tcos\pi/4-cos\omega tsin\pi/4)
2
(sinωtcosπ/4−cosωtsinπ/4)
= \sqrt{2}sin(\omega t-pi/4)
2
sin(ωt−pi/4)
hence, it is simple harmonic motion. and its period = 2π/\omegaω
(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.