When two displacements represented by y_1= a \sin (\omega t)\: \: and \: \: y_2= b \cos (\omega t) are superimposed the motion is:

  • Option 1)

    simple harmonic with amplitude \frac{a}{b}

  • Option 2)

    simple harmonic with amplitude \sqrt{a^{2}+b^{2}}

  • Option 3)

    not a simple harmonic 

  • Option 4)

    simple harmonic with amplitude \frac{a+b}{2}

 

Answers (1)

As we learnt in 

Superposition of sinusoidal wave -

y_{1}= Asin left ( kx-omega t 
ight )\y_{2}= Asin left ( kx-omega t +phi 
ight )

 

- wherein

Resultant equation is y= 2Acdot cos left ( phi /2 
ight )cdot sin left ( kx-omega t+frac{phi }{2} 
ight )

 

 y_{1}=a sinwt

y_{2}=b coswt=bsin(\omega t+\frac{\pi }{2})

y_{eq}=y_{1}+y_{2}

=a sin \omega t + b cos \omega t=a sin \omega t + b sin \left(\omega t+ \frac{\pi}{2}\right )

Since the frequencies for both S.H.M. are same resultant motion will be S.H.M.

A_{eq}=\sqrt{a^{2}+b^{2}+2 ab\ cos\frac{\pi}{2}}

A_{eq}=\sqrt{a^{2}+b^{2}


Option 1)

simple harmonic with amplitude \frac{a}{b}

Incorrect

Option 2)

simple harmonic with amplitude \sqrt{a^{2}+b^{2}}

Correct

Option 3)

not a simple harmonic 

Incorrect

Option 4)

simple harmonic with amplitude \frac{a+b}{2}

Incorrect

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