Phase space deagrams are useful tools .
in analyzing all kinds of dynamical problems. They are especially usrful in studying the changes in motion as initial position and momenum are changed. Here we conseder 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 positon or momentum upwards (or to right) is poitive and downwards (or to left) is negative.
The plase space deagram 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 condetions, and E_(1) and E_(2) the total mechanical energies respectivey.
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Options

A. `E_1 = sqrt(2) E_2`
B. `E_1 = 2 E_2`
C. `E_1 = 4 E_2`
D. `E_1 = 16 E_2`

Answer

C

Explanation

In the 1 st case amplitude of `SHM` is `a`.
In the 2nd case amplitude of `SHM` is `2a`
Total energy `= 1//2 k("amplitude")^(2)`
`E_(1) = (1)/(2) k(2a)^(2) rArr E_(2) = (1)/(2) k(a)^(2)`
`E_(1) = 4E_(2)`.