Question

The displacement of a particle is represented
by the equation
cosWt+cos3 Wt
The motion is
a) non-periodic
b) periodic but not simple harmonic
c) simple harmonic
​

Answer 1

Answer:

Given the equation of displacement of the particle, y=sin3ωt

We know sin3θ=3sinθ−4sin3θ

Hence, y=4(3sinωt−4sin3ωt)⇒4dtdy=3ωcosωt−4×[3ωcos3ωt]⇒4×dt2d2y=−3ω2sinωt+12ωsin3ωt⇒dt2d2y=4−3ω2sinωt+12ωsin3ωt⇒dt2d2y is not proportional to y. 

Hence, the motion is not SHM. 

As the expression is involving sine function, hence it will be periodic. 

Also sin3ωt=(sinωt)3=[sin(ωt+2π)]3=[sin(ωt+2π/ω)]3

Hence, y

SHM

The given function is:

sinωt−cosωt

=2[21sinωt−21cosωt]

=2[sinωt×cos4π−cosωt×sin4π]

=2sin(ωt−4π)

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:

sin3ωt=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[4π−2ωt]

=−3cos[2ωt−4π]

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+c