A mass 1 kg suspended from a spring whose force constant is 400 Nm^-1, executes simple harmonic

step 1 Dear Students, solution for problem 56 of chapter 15 is solution as energy stored is equal to wo

A mass 1 kg suspended from a spring whose force constant is...

A mass $1 \mathrm{~kg}$ suspended from a spring whose force constant is $400 \mathrm{Nm}^{-1}$, executes simple harmonic oscillation. When the total energy of the oscillator is $2 \mathrm{~J}$, the maximum acceleration experienced by the mass will be (a) $2 \mathrm{~ms}^{-2}$ (b) $4 \mathrm{~ms}^{-2}$ (c) $40 \mathrm{~ms}^{-2}$ (d) $400 \mathrm{~ms}^{-2}$

A mass $1 \mathrm{~kg}$ suspended from a spring whose force constant is $400 \mathrm{Nm}^{-1}$, executes simple harmonic oscillation. When the total energy of the oscillator is $2 \mathrm{~J}$, the maximum acceleration experienced by the mass will be
(a) $2 \mathrm{~ms}^{-2}$
(b) $4 \mathrm{~ms}^{-2}$
(c) $40 \mathrm{~ms}^{-2}$
(d) $400 \mathrm{~ms}^{-2}$

Key Concepts

Simple Harmonic Motion

This is a type of periodic oscillatory motion where the restoring force is directly proportional to the displacement from the equilibrium position and acts in the opposite direction. It is a fundamental concept in mechanics, with its behavior described by sine and cosine functions, and is characterized by properties such as frequency, period, amplitude, and phase.

Angular Frequency

Angular frequency, often denoted by ?, quantifies how quickly the system oscillates and is defined as the square root of the ratio of the spring’s force constant to the mass (? = ?(k/m) for a mass-spring system). It plays a crucial role in determining aspects like the period of oscillation and maximum acceleration in simple harmonic motion.

Conservation of Energy in Oscillatory Systems

In a simple harmonic oscillator, the total mechanical energy remains constant, oscillating between kinetic and potential forms. This principle of conservation of energy allows us to relate the energy of the system to its amplitude and to deduce maximum displacement and corresponding maximum values of other dynamical quantities such as acceleration.

Amplitude-Energy Relationship

The energy stored in a simple harmonic oscillator is directly related to the amplitude of oscillation; specifically, the total energy of the system is given by one half the spring constant times the square of the amplitude (E = ½ kA² for mass-spring systems). This relationship is critical for determining the maximum displacement when the energy is known.

Maximum Acceleration in Oscillatory Motion

The maximum acceleration experienced by an object in simple harmonic motion occurs when it is at its maximum displacement and is calculated using the relation a_max = ?²A. This link between angular frequency and amplitude is essential for understanding the dynamics of oscillatory systems and for computing various instantaneous quantities.

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