When a block of mass m is suspended from the free end of a...

When a block of mass $\mathrm{m}$ is suspended from the free end of a massless spring having force constant $\mathrm{k}$, its length increases by y. Now when the block is slightly pulled downwards and released, it starts executing S.H.M with amplitude $\mathrm{A}$ and angular frequency $\omega$. The total energy of the system comprising of the block and spring is $\ldots \ldots \ldots$ (A) $(1 / 2) \mathrm{m} \omega^{2} \mathrm{~A}^{2}$ (B) $(1 / 2) m \omega^{2} A^{2}+(1 / 2) \mathrm{ky}^{2}$ (C) $(1 / 2) \mathrm{ky}^{2}$ (D) $(1 / 2) m \omega^{2} A^{2}-(1 / 2) k y^{2}$

When a block of mass $\mathrm{m}$ is suspended from the free end of a massless spring having force constant $\mathrm{k}$, its length increases by y. Now when the block is slightly pulled downwards and released, it starts executing S.H.M with amplitude $\mathrm{A}$ and angular frequency $\omega$. The total energy of the system comprising of the block and spring is $\ldots \ldots \ldots$
(A) $(1 / 2) \mathrm{m} \omega^{2} \mathrm{~A}^{2}$
(B) $(1 / 2) m \omega^{2} A^{2}+(1 / 2) \mathrm{ky}^{2}$
(C) $(1 / 2) \mathrm{ky}^{2}$
(D) $(1 / 2) m \omega^{2} A^{2}-(1 / 2) k y^{2}$

Key Concepts

Simple Harmonic Motion

Simple harmonic motion (SHM) describes an oscillatory movement where the restoring force acting on the object is directly proportional to its displacement from an equilibrium position and is directed toward that equilibrium position. This concept is fundamental for analyzing the motion of the spring-block system after it is set into oscillation.

Equilibrium in a Spring-Mass System

In a spring-mass system, the static equilibrium position is attained when the downward force due to gravity is exactly balanced by the upward force provided by the spring (Hooke's law). Mathematically, this equilibrium condition is expressed as mg = k * y, where y is the displacement of the spring from its natural length, defining the baseline about which oscillations occur.

Energy in Simple Harmonic Motion

The total mechanical energy in simple harmonic motion is conserved and is comprised of a combination of kinetic energy and potential energy from the spring. Because oscillations occur around the equilibrium position, the gravitational potential energy (which has been balanced out at equilibrium) does not contribute to the energy variations during oscillation, leading to the energy being expressed solely in terms of the oscillatory kinetic and spring potential energies.

Angular Frequency and its Relation to System Parameters

The angular frequency of a spring-mass oscillator is given by the relation ?² = k/m, linking the physical properties of the spring (its stiffness, k) and the mass attached to it. This relationship is essential to understanding the timing and energy of oscillations, as the total energy in SHM is traditionally written in terms of ? and the amplitude of motion.

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