Given, displacement
y=0.2sin(10πt+1.5π)cos(10πt+1.5π)
Considering (10πt+1.5π)=θ
∴y=0.2sinθcosθ
Multiplying and dividing by 2,
y=0.1×sin2θ (∵2sinθcosθ=sin2θ)
Putting value of θ back,
y=0.1×sin2(10πt+1.5π)
=0.1sin(20πt+3π)
Comparing with standard equation of SHM y=Asin(ωt+ϕ),
we get, ω=20π
Time period is given as T=2π/ω
⇒T=2π20π
⇒T=0.1 s
Hence, motion of the particle is simple harmonic with time period =0.1 s