A body of mass 1 kg suspended from the free end of a spring...

A body of mass $1 \mathrm{~kg}$ suspended from the free end of a spring having force constant $400 \mathrm{Nm}^{-1}$ is executing S.H.M. When the total energy of the system is 2 joule, the maximum acceleration is $\ldots \ldots . \mathrm{ms}^{-2}$. (A) $8 \mathrm{~ms}^{-2}$ (B) $10 \mathrm{~ms}^{-2}$ (C) $40 \mathrm{~ms}^{-2}$ (D) $40 \mathrm{cms}^{-2}$

A body of mass $1 \mathrm{~kg}$ suspended from the free end of a spring having force constant $400 \mathrm{Nm}^{-1}$ is executing S.H.M. When the total energy of the system is 2 joule, the maximum acceleration is $\ldots \ldots . \mathrm{ms}^{-2}$.
(A) $8 \mathrm{~ms}^{-2}$
(B) $10 \mathrm{~ms}^{-2}$
(C) $40 \mathrm{~ms}^{-2}$
(D) $40 \mathrm{cms}^{-2}$

Key Concepts

Simple Harmonic Motion (SHM)

SHM is a type of periodic motion where the restoring force is directly proportional to the displacement from an equilibrium position and acts in the opposite direction. It is characterized by sinusoidal oscillations, constant angular frequency, and the transfer of energy between kinetic and potential forms.

Conservation of Energy in Oscillatory Systems

In oscillatory systems like a mass-spring system, the total mechanical energy (sum of kinetic and potential energy) remains constant when no non-conservative forces act. At maximum displacement, the system's energy is stored entirely as potential energy, while at the equilibrium position, it is entirely kinetic.

Relation Between Amplitude and Energy

For a mass-spring system, the total energy is given by E = 1/2 k A^2, where k is the spring constant and A is the amplitude of oscillation. This relationship allows one to calculate the amplitude when the total energy and spring constant are known.

Angular Frequency

Angular frequency, denoted by ?, is a measure of how quickly an object oscillates in SHM and is defined by the relationship ? = ?(k/m), where k is the spring constant and m is the mass. It is crucial in linking the system's properties to its time-dependent behavior.

Maximum Acceleration in SHM

The maximum acceleration in SHM occurs at the maximum displacement and can be calculated using the formula a_max = ?^2 A, where ? is the angular frequency and A is the amplitude. This formula directly relates the system's physical parameters to the extreme value of acceleration experienced during the motion.

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