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← IIT-JEE 2011 · Paper 1
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IIT-JEE 2011 · Paper 1, Physics question 36

Passage

Paragraph for Question Nos. 35 to 37

Phase space diagrams are useful tools in analyzing all kinds of dynamical problems. They are especially useful in studying the changes in motion as initial position and momentum are changed. Here we consider some simple dynamical systems in one-dimension. For such systems, phase space is a plane in which position is plotted along horizontal axis and momentum is plotted along vertical axis. The phase space diagram is x(t) vs. p(t) curve in this plane. The arrow on the curve indicates the time flow. For example, the phase space diagram for a particle moving with constant velocity is a straight line as shown in the figure. We use the sign convention in which position or momentum upwards (or to right) is positive and downwards (or to left) is negative.

Momentum against position: a horizontal line at a positive momentum with an arrow pointing to the right (constant velocity).

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 E1E_1 and E2E_2 are the total mechanical energies respectively. Then

Two concentric circles about the origin of the position-momentum plane, both traversed clockwise: the inner one (E₂) of radius a and the outer one (E₁) of radius 2a.
Choose one option
Answer and worked solution

Official answer: C

Energy and amplitude. For the same oscillator E=12kA2E=\frac12kA^2; the outer circle has amplitude 2a2a, the inner aa:
E1E2=(2aa)2=4.\frac{E_1}{E_2}=\left(\frac{2a}a\right)^2=4.

(C)