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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 get
a2+b2=A2cos2ϕ+A2sin2ϕ=A2
A=a2+b2
Now, y=Acosϕ.sinωt+Asinϕcosωt
y=Asin(ωt+ϕ)
dydt=Aωcos(ωt+ϕ)
d2ydt2=Aω2sin(ωt+ϕ)
d2ydt2=Aω2y
d2ydt2(y)
Hence, the given equation is S.H.M with amplitude a2+b2
The correct option is D.

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