Which of the following function of time represents simple harmonic motion
i. sin wt - cos wt
ii. sin² wt
iii. cos wt + cos 3 wt + cos 5 wt
iv. 1 + wt + w²t²
Answers
Answer:
(a) SHM
The given function is:
sinωt−cosωt
=
2
[
2
1
sinωt−
2
1
cosωt]
=
2
[sinωt×cos
4
π
−cosωt×sin
4
π
]
=
2
sin(ωt−
4
π
)
This function represents SHM as it can be written in the form: a sin(ωt+ϕ) Its period is: 2π/ω.
(b) Periodic but not SHM
The given function is:
sin
3
ωt=1/4[3sinωt−sin3ωt]
The terms sin ωt and sin 3ωt individually represent simple harmonic motion (SHM). However, the superposition of two SHM is periodic and not simple harmonic.
Its period is: 2π/ω (LCM of time periods).
(c) SHM
The given function is:
3cos[
4
π
−2ωt]
=−3cos[2ωt−
4
π
]
This function represents simple harmonic motion because it can be written in the form: a cos(ωt+ϕ). Its period is: 2π/2ω=π/ω
(d) Periodic, but not SHM
The given function is cosωt+cos3ωt+cos5ωt. Each individual cosine function represents SHM. However, the superposition of three simple harmonic motions is periodic, but not simple harmonic.
Its period is LCM of the period of the three sinusoids = 2π/ω