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Question

The displacement y of a particle varies with time t according to the relation y=asinωt+bcosωt. Which of the following statement(s) is/are correct regarding the motion?

A
The motion is SHM.
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B
The motion is SHM with amplitude a+b.
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C
The motion is SHM with amplitude a2+b2.
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D
The motion is SHM with amplitude a2+b2.
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Solution

The correct option is D The motion is SHM with amplitude a2+b2.
From the data given in the question, the displacement
y=asinωt+bcosωt ....(1)
Let a=Acosϕ and b=Asinϕ
Now, a2+b2=A2cos2ϕ+A2sin2ϕ=A2
A=a2+b2
From (1), we can say that
y=Acosϕ.sinωt+Asinϕ.cosωt=Asin(ωt+ϕ)

Double differentiating with respect to time on both sides, we get
dydt=Aωcos(ωt+ϕ)
d2ydt2=Aω2sin(ωt+ϕ)=ω2y
d2ydt2(y)
Hence, it is the equation of an SHM with amplitude A=a2+b2

Hence, options (a) and (d) are correct.

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