A system is executing S.H.M. The potential energy of the...

A system is executing S.H.M. The potential energy of the system for displacement $\mathrm{x}$ is $\mathrm{E}_{1}$ and for a displacement of $\mathrm{y}$, the potential energy of the system is $E_{2}$. The potential energy for a displacement of $(x+j)$ is $\ldots \ldots \ldots$ (A) $E_{1}+E_{2}$ (B) $\left.\sqrt{(} \mathrm{E}_{1}^{2}+\mathrm{E}_{2}{ }^{2}\right)$ (C) $\mathrm{E}_{1}+\mathrm{E}_{2}+2 \sqrt{\mathrm{E}}_{1} \mathrm{E}_{2}$ (D) $\sqrt{E}_{1} E_{2}$

A system is executing S.H.M. The potential energy of the system for displacement $\mathrm{x}$ is $\mathrm{E}_{1}$ and for a displacement of $\mathrm{y}$, the potential energy of the system is $E_{2}$. The potential energy for a displacement of $(x+j)$ is $\ldots \ldots \ldots$
(A) $E_{1}+E_{2}$
(B) $\left.\sqrt{(} \mathrm{E}_{1}^{2}+\mathrm{E}_{2}{ }^{2}\right)$
(C) $\mathrm{E}_{1}+\mathrm{E}_{2}+2 \sqrt{\mathrm{E}}_{1} \mathrm{E}_{2}$
(D) $\sqrt{E}_{1} E_{2}$

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Key Concepts

Simple Harmonic Motion (SHM)

Simple Harmonic Motion describes a type of periodic oscillation where the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction. This concept is foundational in understanding many physical systems, such as springs and pendulums, and serves as a model for analyzing oscillatory motion in terms of energy and forces.

Quadratic Dependence of Potential Energy

In systems exhibiting SHM, the potential energy is typically a quadratic function of displacement. For example, in a spring, the potential energy is given by (1/2)kx², where x is the displacement. This quadratic relationship is significant because when combining displacements, the potential energy does not simply add linearly; instead, the combination leads to additional cross terms, as demonstrated by the expansion of the square of a sum.

Binomial Expansion in Energy Calculations

When combining displacements in a harmonic oscillator, the total potential energy for a displacement that is the sum of two separate displacements is determined by the binomial expansion. This means the energy includes not only the sum of individual energy contributions but also a term that represents the interaction of these displacements—a feature that arises naturally from the quadratic form of the potential energy.

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