Two simple harmonic motions are given by x1-a2sin- t - -3a2cos- t and x2 -asin- t - acos- t- What is the ratio of the amplitude of the first motion to that of the second -

Amplitude of the first simple harmonic motion is A_1 = √( (a2 nbsp; )^2 + ( √(3)a2 )^2)=a Amplitude of the second motion is A_2=√(a^2+a^2)=√(2)a ∴ nbsp; A ...

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Standard XII

Physics

SHM:The Calculus Picture

Two simple ha...

Question

Two simple harmonic motions are given by $x_{1} = \frac{a}{2} sin ω t + \frac{\sqrt{3} a}{2} cos ω t$ and $x_{2} = a sin ω t + a cos ω t$ . What is the ratio of the amplitude of the first motion to that of the second ?

\frac{\sqrt{3}}{2}

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\frac{1}{\sqrt{2}}

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\sqrt{2}

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\frac{1}{2}

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Solution

The correct option is B $\frac{1}{\sqrt{2}}$
Amplitude of the first simple harmonic motion is
$A_{1} = \sqrt{{(\frac{a}{2})}^{2} + {(\frac{\sqrt{3} a}{2})}^{2}} = a$
Amplitude of the second motion is
$A_{2} = \sqrt{a^{2} + a^{2}} = \sqrt{2} a$
$∴$ $\frac{A_{1}}{A_{2}} = \frac{1}{\sqrt{2}}$
Hence, the correct choice is (b)

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Two simple harmonic motions are given by x1=a2sinωt+√3a2cosωt and x2=asinωt+acosωt. What is the ratio of the amplitude of the first motion to that of the second ?

The correct option is B 1√2Amplitude of the first simple harmonic motion is A1= ⎷(a2)2+(√3a2)2=a Amplitude of the second motion is A2=√a2+a2=√2a ∴ A1A2=1√2 Hence, the correct choice is (b)

Two simple harmonic motions are given by $x_{1} = \frac{a}{2} sin ω t + \frac{\sqrt{3} a}{2} cos ω t$ and $x_{2} = a sin ω t + a cos ω t$ . What is the ratio of the amplitude of the first motion to that of the second ?