The displacement of a particle executing simple harmonic motion is given by, Then the amplitude of its oscillation is given by [2019]

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Q.
The displacement of a particle executing simple harmonic motion is given by, $y = A_{0} + A \sin ω t + B \cos ω t .$ Then the amplitude of its oscillation is given by [2019]

1 $A + B$

2 $A_{0} + \sqrt{A^{2} + B^{2}}$

3 $\sqrt{A^{2} + B^{2}}$

4 $\sqrt{A_{0}^{2} + {(A + B)}^{2}}$

Ans.
(3)

$y = A_{0} + A \sin ω t + B \cos ω t$

or $(y - A_{0}) = A \sin ω t + B \cos ω t$ or $y' = A \sin ω t + B \cos ω t$

$= A \cos (\frac{π}{2} - ω t) + B \cos ω t$

Amplitude $= \sqrt{A^{2} + B^{2} + 2 A B \cos \frac{π}{2}}$ $[∵ ϕ = \frac{π}{2}]$

$= \sqrt{A^{2} + B^{2}}$

Solution

Q.
The displacement of a particle executing simple harmonic motion is given by, $y = A_{0} + A \sin ω t + B \cos ω t .$ Then the amplitude of its oscillation is given by [2019]

1 $A + B$

2 $A_{0} + \sqrt{A^{2} + B^{2}}$

3 $\sqrt{A^{2} + B^{2}}$

4 $\sqrt{A_{0}^{2} + {(A + B)}^{2}}$

Ans.
(3)

$y = A_{0} + A \sin ω t + B \cos ω t$

or $(y - A_{0}) = A \sin ω t + B \cos ω t$ or $y' = A \sin ω t + B \cos ω t$

$= A \cos (\frac{π}{2} - ω t) + B \cos ω t$

Amplitude $= \sqrt{A^{2} + B^{2} + 2 A B \cos \frac{π}{2}}$ $[∵ ϕ = \frac{π}{2}]$

$= \sqrt{A^{2} + B^{2}}$

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Solution

Q. The displacement of a particle executing simple harmonic motion is given by, y=A0+Asinωt+Bcosωt. Then the amplitude of its oscillation is given by [2019]

1 A+B

2 A0+A2+B2

3 A2+B2

4 A02+(A+B)2

Ans. (3) y=A0+Asinωt+Bcosωt or (y-A0)=Asinωt+Bcosωt or y'=Asinωt+Bcosωt =Acos(π2-ωt)+Bcosωt Amplitude =A2+B2+2ABcosπ2 [∵ϕ=π2] =A2+B2

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Q.
The displacement of a particle executing simple harmonic motion is given by, $y = A_{0} + A \sin ω t + B \cos ω t .$ Then the amplitude of its oscillation is given by [2019]

1 $A + B$

2 $A_{0} + \sqrt{A^{2} + B^{2}}$

3 $\sqrt{A^{2} + B^{2}}$

4 $\sqrt{A_{0}^{2} + {(A + B)}^{2}}$

Ans.
(3)

$y = A_{0} + A \sin ω t + B \cos ω t$

or $(y - A_{0}) = A \sin ω t + B \cos ω t$ or $y' = A \sin ω t + B \cos ω t$

$= A \cos (\frac{π}{2} - ω t) + B \cos ω t$

Amplitude $= \sqrt{A^{2} + B^{2} + 2 A B \cos \frac{π}{2}}$ $[∵ ϕ = \frac{π}{2}]$

$= \sqrt{A^{2} + B^{2}}$