One end of a mass less spring having force constant k and length 50 cm is attached at the upper

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One end of a mass less spring having force constant k and...

One end of a mass less spring having force constant $\mathrm{k}$ and length $50 \mathrm{~cm}$ is attached at the upper end of a plane inclined at an angle $e=30^{\circ} .$ When a body of mass $m=1.5 \mathrm{~kg}$ is attached at the lower end of the spring, the length of the spring increases by $2.5 \mathrm{~cm}$. Now, if the mass is displaced by a small amount and released, the amplitude of the resultant oscillation is ......... (A) $(\pi / 7)$ (B) $(2 \pi / 7)$ (C) $(\pi / 5)$ (D) $(2 \pi / 5)$

One end of a mass less spring having force constant $\mathrm{k}$ and length $50 \mathrm{~cm}$ is attached at the upper end of a plane inclined at an angle $e=30^{\circ} .$ When a body of mass $m=1.5 \mathrm{~kg}$ is attached at the lower end of the spring, the length of the spring increases by $2.5 \mathrm{~cm}$. Now, if the mass is displaced by a small amount and released, the amplitude of the resultant oscillation is .........
(A) $(\pi / 7)$
(B) $(2 \pi / 7)$
(C) $(\pi / 5)$
(D) $(2 \pi / 5)$

Key Concepts

Hooke's Law

Hooke’s law states that the restoring force exerted by an elastic spring is directly proportional to the displacement from its natural length, acting in the opposite direction. This principle is fundamental in understanding how springs behave under load, and it provides the basis for modeling the force in mass?spring systems and their oscillatory motion.

Static Equilibrium in a Gravitational Field

When a spring?mass system is subject to gravity, the mass will stretch the spring until the restoring force of the spring balances the component of the gravitational force acting along the direction of the spring. This condition of static equilibrium, reached with a certain extension of the spring, is key to determining the system’s new equilibrium position before any dynamic effects (such as oscillations) take place.

Simple Harmonic Motion (SHM)

When a mass attached to a spring is displaced slightly from its equilibrium position and released, it experiences a restoring force that is proportional to its displacement. Under the assumption of small oscillations, this leads to simple harmonic motion, where the system oscillates sinusoidally with an angular frequency that depends on the mass and the spring constant.

Small Oscillations About Equilibrium

In the context of systems that are already offset by a static displacement, small additional displacements result in oscillations about the equilibrium position. The amplitude of these oscillations is determined by the initial displacement from equilibrium and remains independent of the constant gravitational force, because gravity merely shifts the equilibrium position without affecting the oscillatory dynamics governed by the spring’s restoring force.

Transcript

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