As shown in figure, a spring attached to the ground...

As shown in figure, a spring attached to the ground vertically has a horizontal massless plate with a $2 \mathrm{~kg}$ mass in it. When the spring (massless) is pressed slightly and released, the $2 \mathrm{~kg}$ mass, starts executing S.H.M. The force constant of the spring is $200 \mathrm{Nm}^{-1}$. For what minimum value of amplitude, will the mass loose contact with the plate? (Take $\left.\mathrm{g}=10 \mathrm{~ms}^{-2}\right)$ (A) $10.0 \mathrm{~cm}$ (B) $8.0 \mathrm{~cm}$ (C) $4.0 \mathrm{~cm}$ (D) For any value less than $12.0 \mathrm{~cm}$.

As shown in figure, a spring attached to the ground vertically has a horizontal massless plate with a $2 \mathrm{~kg}$ mass in it. When the spring (massless) is pressed slightly and released, the $2 \mathrm{~kg}$ mass, starts executing S.H.M. The force constant of the spring is $200 \mathrm{Nm}^{-1}$. For what minimum value of amplitude, will the mass loose contact with the plate? (Take $\left.\mathrm{g}=10 \mathrm{~ms}^{-2}\right)$
(A) $10.0 \mathrm{~cm}$
(B) $8.0 \mathrm{~cm}$
(C) $4.0 \mathrm{~cm}$
(D) For any value less than $12.0 \mathrm{~cm}$.

Key Concepts

Simple Harmonic Motion (SHM)

SHM describes a type of periodic motion where the restoring force is directly proportional to the displacement from equilibrium. In the context of oscillating systems like a spring-mass arrangement, the motion can be modeled with sinusoidal functions, and key parameters such as amplitude, angular frequency, and phase determine the dynamics of the system.

Natural Frequency of a Spring-Mass System

The natural frequency is a crucial characteristic of a spring-mass system and is defined by the relationship ? = ?(k/m), where k is the spring constant and m is the mass. This frequency determines how fast the system oscillates and plays an important role in analyzing the acceleration experienced during SHM.

Normal Reaction Force and Loss of Contact

When an object is in contact with an oscillating platform, the contact is maintained by the normal reaction force. For vertical oscillations, if the acceleration of the platform becomes greater in magnitude than gravitational acceleration (g) in the downward direction, the normal force can vanish, causing the object to lose contact. The condition for losing contact is found by equating the maximum acceleration of the platform with g, which helps determine the minimum amplitude needed for detachment.

Force Analysis in an Accelerating System

A proper force analysis involves considering all forces acting on the object, including gravitational force and the inertial effects due to the acceleration of the platform. By setting the net force (which includes the spring-induced acceleration and gravity) to zero at the point of imminent separation, one can derive conditions that determine when the mass will lose contact, linking the kinematics of SHM with dynamic constraints.

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