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Standard XII

Physics

Velocity Displacement Relationship

The phase spa...

Question

The phase space diagram for simple harmonic motion is a circle centered at the origin. In the figure, the two circles represent the same oscillator but for different initial conditions, and $E_{1}$ and $E_{2}$ are the total mechanical energies respectively. Then

E_{1} = \sqrt{2} E_{2}

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E_{1} = 2 E_{2}

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E_{1} = 4 E_{2}

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E_{1} = 16 E_{2}

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Solution

The correct option is B $E_{1} = 4 E_{2}$
For, a S.H.M the total mechanical energy $E \propto A^{2}$ ( $A$ =amplitude)
In case of small circle amplitude $a$ and energy $E_{2}$ and in case of large circle amplitude is $2 a$ and energy $E_{1}$ .
So, $\frac{E_{2}}{E_{1}} = \frac{{(a)}^{2}}{{(2 a)}^{2}}$
$\Rightarrow E_{1} = 4 E_{2}$
$*$ N.B: In case of circular motion the radius of circle donates the amplitude of S.H.M when it is projected along its diameter.

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The phase space diagram for simple harmonic motion is a circle centered at the origin. In the figure, the two circles represent the same oscillator but for different initial conditions, and E1 and E2 are the total mechanical energies respectively. Then

The phase space diagram for simple harmonic motion is a circle centered at the origin. In the figure, the two circles represent the same oscillator but for different initial conditions, and $E_{1}$ and $E_{2}$ are the total mechanical energies respectively. Then