Which of the following examples represent periodic motion?a. A swimming completing one (return) trip from one bank of a river to the other and back.b. A freely suspended bar magnet displaced from its N-S direction and released.c. A hydrogen molecule rotating about its centre of mass.d. An row released from a bow.
Answers
Let's check which one is period or not ,
(a) Here, the motion of the swimmer between the banks of the river is to and fro. But it cannot be said to have a definite period as we do not know how much time the swimmer will need to complete each trip. Hence, it does not represent a periodic motion.
(b) When a freely suspended magnet is displaced from N-S direction and released, it oscillates periodically about the mean position. Hence, it is a periodic motion.
(c) The hydrogen molecule rotates about its centre of mass which constitutes a periodic motion.
(d) When an arrow is released from a bow, it just goes in one direction and doesn’t make oscillations to and fro . Hence, the motion is non-periodic.
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.