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Simple Harmonic Motion question

2011 · Shift 1 · Q65
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Simple Harmonic Motion question

2011 · Shift 1 · Q65

JEE AdvancedPhysicsSimple Harmonic MotionMCQ+3 / −1
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 int he figure. We use the sign convention in which position or momentum upwards (or to right) is positive and downwards (or to left) is negative. IIT-JEE 2011 Paper 1 Offline Physics - Simple Harmonic Motion Question 16 English ComprehensionConsider the spring-mass system, with the mass submerged in water, as shown in the figure. The phase space diagram for one cycle of this system is IIT-JEE 2011 Paper 1 Offline Physics - Simple Harmonic Motion Question 16 English
  1. A
    IIT-JEE 2011 Paper 1 Offline Physics - Simple Harmonic Motion Question 16 English Option 1
  2. B
    IIT-JEE 2011 Paper 1 Offline Physics - Simple Harmonic Motion Question 16 English Option 2
  3. C
    IIT-JEE 2011 Paper 1 Offline Physics - Simple Harmonic Motion Question 16 English Option 3
  4. D
    IIT-JEE 2011 Paper 1 Offline Physics - Simple Harmonic Motion Question 16 English Option 4
View written solutionFree

Correct answer: B

  1. Identify the system

    A spring-mass system submerged in water experiences:

    • restoring force of spring: Fs=−kxF_s=-kxFs​=−kx
    • damping force due to water: Fd=−bvF_d=-bvFd​=−bv

    So the equation of motion is mx¨+bx˙+kx=0m\ddot x+b\dot x+kx=0mx¨+bx˙+kx=0

    This is a damped harmonic oscillator.

  2. Relate phase space variables

    In phase space, we plot:

    • horizontal axis: position xxx
    • vertical axis: momentum p=mx˙p=m\dot xp=mx˙

    For an undamped SHM, the trajectory is a closed ellipse in the xxx-ppp plane.

    But here damping is present, so mechanical energy continuously decreases with time.

  3. Effect of damping on phase trajectory

    Since energy decreases, both amplitude of oscillation and maximum momentum decrease with time.

    Therefore, instead of a closed ellipse, the phase point spirals inward toward the origin: (x,p)→(0,0)(x,p) \to (0,0)(x,p)→(0,0)

    The origin corresponds to the mass finally coming to rest at equilibrium.

  4. Direction of motion in phase space

    The motion still oscillates while gradually losing amplitude, so the trajectory winds around the origin repeatedly, but with shrinking size.

    Hence the correct phase-space plot must be an inward spiral.

  5. Choose the correct option

    Among the given options, the one representing an inward spiral toward the origin is Option B.

  6. Comparison with stored answer

    Derived answer: B

    Stored correct answer: B

    They match.

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