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Question

# The displacement of a particle varies with time according to the relation y=asinωt+bcosωt

A
The motion is oscillatory but not S.H.M.
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B
The motion is S.H.M. with amplitude a+b.
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C
The motion is S.H.M. with amplitude a2+b2.
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D
The motion is S.H.M. with amplitude a2+b2.
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Solution

## The correct option is D The motion is S.H.M. with amplitude √a2+b2.According to the question, the displacement,y=asinωt+bcosωt. . . . . . . .(1)let, a=Acosϕ. . . . . .(2) b=Asinϕ.. . . . .(3)Squaring and adding equation (2) and (3), we geta2+b2=A2cos2ϕ+A2sin2ϕ=A2A=√a2+b2Now, y=Acosϕ.sinωt+Asinϕcosωty=Asin(ωt+ϕ)dydt=Aωcos(ωt+ϕ)d2ydt2=−Aω2sin(ωt+ϕ)d2ydt2=−Aω2yd2ydt2∝(−y)Hence, the given equation is S.H.M with amplitude √a2+b2The correct option is D.

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