As shown in figure, a block A having mass M is attached to...

As shown in figure, a block A having mass $M$ is attached to one end of a massless spring. The block is on a frictionless horizontal surface and the free end of the spring is attached to a wall. Another block B having mass ' $\mathrm{m}$ ' is placed on top of block A. Now on displacing this system horizontally and released, it executes S.H.M. What should be the maximum amplitude of oscillation so that B does not slide off A? Coefficient of static friction between the surfaces of the block's is $\mu$. (A) $A_{\max }=\{(\mu \mathrm{mg}) / \mathrm{k}\}$ (B) $A_{\max }=[\{\mu(m+M) g\} / k]$ (C) $A_{\max }=[\{\mu(M-\mathrm{m}) g\} / \mathrm{k}]$ (D) $A_{\max }=[\{2 \mu(M+m)\} / k]$

As shown in figure, a block A having mass $M$ is attached to one end of a massless spring. The block is on a frictionless horizontal surface and the free end of the spring is attached to a wall. Another block B having mass ' $\mathrm{m}$ ' is placed on top of block A. Now on displacing this system horizontally and released, it executes S.H.M. What should be the maximum amplitude of oscillation so that B does not slide off A? Coefficient of static friction between the surfaces of the block's is $\mu$.
(A) $A_{\max }=\{(\mu \mathrm{mg}) / \mathrm{k}\}$
(B) $A_{\max }=[\{\mu(m+M) g\} / k]$
(C) $A_{\max }=[\{\mu(M-\mathrm{m}) g\} / \mathrm{k}]$
(D) $A_{\max }=[\{2 \mu(M+m)\} / k]$

Key Concepts

Simple Harmonic Motion

Simple Harmonic Motion (SHM) refers to a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that displacement. This concept is key in understanding systems where acceleration and displacement are linearly related, typically described by the equation a = - (k/m_eff)x, where k is the spring constant and m_eff is the effective mass of the oscillating system.

Static Friction

Static friction is the force that must be overcome to initiate relative motion between two surfaces in contact. It is quantified by the product of the coefficient of static friction (?) and the normal force. This concept is crucial when determining conditions under which one object, resting on another, remains stationary relative to the moving base, as it sets the maximum force that can be applied without slipping.

Relative Motion in Accelerating Systems

Relative motion in accelerating systems considers the inertial effects when one body is moving relative to another that is accelerating. In contexts where friction is used to transmit acceleration between bodies (such as one block on top of another), analyzing the relative acceleration helps determine whether the frictional force is sufficient to keep the bodies moving together without slipping.

Effective Mass in Oscillatory Systems

When two or more masses move together in an oscillatory motion, the system behaves as if it possesses a single effective mass. This concept simplifies the analysis by allowing the application of SHM principles, where the effective mass is the sum (or an appropriate combination) of the individual masses participating in the motion, thereby affecting both the system’s acceleration and the oscillation frequency.

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