Question

Both the equations y1 = a sin omega t and y2= a sin omega t + a/2 cos omega t represent shm the ratio of amplitude of the two motions is

Answer 1

ratio of amplitudes of the two motions is 2 : √5

given, y1 = a sinωt and y2 = a sinωt + a/2cosωt

as it is clear that amplitude of wave y1 is a but amplitude of wave y2 isn't clear let's resolve 2nd equation of wave.

y2 = a sinωt + a/2cosωt

= a/2 [ 2sinωt + cosωt ]

= √5a/2 [ 2/√5 sinωt + 1/√5 cosωt ]

let cosα = 2/√5 then, sinα = 1/√5

= √5a/2 [ cosα sinωt + sinα cosωt ]

= √5a/2 sin(ωt + α)

so, amplitude of wave y2 is √5a/2 .

so, ratio of amplitudes of the two motions is a/(√5a/2) = 2/√5

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