A block attached to an ideal spring undergoes simple...

A block attached to an ideal spring undergoes simple harmonic motion about its equilibrium position $(x=0)$ with amplitude $A$ . What fraction of the total energy is in the form of kinetic energy when the block is at position $x=\frac{1}{2} A ?$ (A) $\frac{1}{3}$ (B) $\frac{1}{2}$ (C) $\frac{2}{3}$ (D) $\frac{3}{4}$

A block attached to an ideal spring undergoes simple harmonic motion about its equilibrium position $(x=0)$ with amplitude $A$ . What fraction of the total energy is in the form of kinetic energy when the block is at position $x=\frac{1}{2} A ?$
(A) $\frac{1}{3}$
(B) $\frac{1}{2}$
(C) $\frac{2}{3}$
(D) $\frac{3}{4}$

Key Concepts

Simple Harmonic Motion

Simple harmonic motion (SHM) describes periodic motion where the restoring force is directly proportional to the displacement and acts in the opposite direction. In the context of a block attached to an ideal spring, SHM provides the framework to analyze the motion by relating displacement, velocity, and acceleration through sinusoidal functions.

Energy Conservation in a Spring-Mass System

In a mass-spring system undergoing SHM, the total mechanical energy remains constant if no non-conservative forces act. This total energy is the sum of kinetic energy and potential energy, and it is conserved throughout the oscillation, allowing us to relate the energy at different positions of the oscillating mass.

Potential Energy in an Elastic Spring

The potential energy stored in a spring, when it is displaced from its equilibrium position, is given by (1/2)kx², where k is the spring constant and x is the displacement. This quadratic relationship is key to understanding how energy is stored in the spring and how it changes as the mass moves.

Kinetic Energy in SHM and Energy Partitioning

At any given position in SHM, the kinetic energy of the mass can be determined by subtracting the potential energy from the total energy (which is (1/2)kA² for a maximum amplitude A). This partitioning of energy into kinetic and potential forms highlights that as the mass moves away from the equilibrium position, potential energy increases while kinetic energy correspondingly decreases, maintaining overall energy conservation.

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