The displacement of a particle varies with time according to the relation, \(y=asin \omega t+bcos \omega t.\)

1. the motion is oscillatory but not SHM.
2. the motion is SHM with amplitude \(a+b.\)
3. the motion is SHM with amplitude \(a^{2}+b^{2}.\)
4. the motion is SHM with amplitude \(\sqrt{a^{2}+b^{2}}\).
4. Hint: Apply the concept of the superposition principle.
Step 1: Find the combined equation of the motion.
According to the question, the displacement,
y=a sinωt+bcosωt
Let a=Asinϕ and b=Acosϕ
Now, a2+b2=A2sin2ϕ+A2cos2ϕ=A2A=a2+b2
y=A sinϕsinωt+A cosϕ cosωt=A sin (ωt+ϕ)
Step 2: Find the acceleration of the motion.
dydt=Aω cos(ωt+ϕ)
d2ydt2=-Aω2sin(ωt+ϕ)=-Ayω2=(-Aω2)y
       d2ydt2(-y)
Hence, it is an equation of SHM with amplitude A=a2+b2