Physics

Problem

(Mass–spring system under general initial conditions) a. (i) Define amplitude and phase constant for the linear simple harmonic motion of a particle attached to an ideal spring. State the physical significance of each term. b. (ii) A particle executing SHM has an initial displacement

x_{0}

and initial velocity

v_{0}

t=0

. Explain qualitatively, without using any numerical values, why knowledge of both

x_{0}

and

v_{0}

is sufficient to fix the amplitude and the phase constant uniquely. c. (iii) Using suitable reasoning, identify the value(s) of the phase constant that correspond to the following special initial situations: (a) the particle is released from one extreme position, (b) the particle passes through the mean position moving in the positive

x

–direction, (c) the particle is momentarily at rest at the mean position.

Intuitive explanation

When a mass bounces on a spring, its motion is completely determined by two things: where it starts and how fast it's moving at that moment. The key insight is that any oscillation can be built from a combination of two independent modes—one that depends on initial position and one that depends on initial velocity. By writing the solution as $x(t) = A\cos(\omega t + \phi)$ , you're really asking: what amplitude and phase angle make this formula match your specific starting conditions? Use those two initial conditions to solve for those two unknowns.

Intuitive explanation

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