A block is placed on a horizontal table. The table executes...

A block is placed on a horizontal table. The table executes S.H.M. along the horizontal plane with a period $\mathrm{T}$. The coefficient of static friction between the table and block is $\mu$. The maximum amplitude of oscillation should be $\ldots \ldots$.. so that the block does not slide off the table. (A) $(\mu \mathrm{g} \mathrm{T} / 5 \pi)$ (B) $\left\{\left(\mu \mathrm{g} \mathrm{T}^{2}\right) / 4 \pi^{2}\right\}$ (C) $\{(\mu \mathrm{gT}) / 2 \pi\}$ (D) $\mu \mathrm{gT}$

A block is placed on a horizontal table. The table executes
S.H.M. along the horizontal plane with a period $\mathrm{T}$. The coefficient of static friction between the table and block is $\mu$. The maximum amplitude of oscillation should be $\ldots \ldots$.. so that the block does not slide off the table.
(A) $(\mu \mathrm{g} \mathrm{T} / 5 \pi)$
(B) $\left\{\left(\mu \mathrm{g} \mathrm{T}^{2}\right) / 4 \pi^{2}\right\}$
(C) $\{(\mu \mathrm{gT}) / 2 \pi\}$
(D) $\mu \mathrm{gT}$

Key Concepts

Static Friction

Static friction is the force that resists the initiation of sliding motion between two surfaces in contact. Its maximum value is proportional to the normal force, given by the product of the coefficient of static friction and the normal reaction (usually mg on a horizontal surface). This concept is crucial when analyzing the conditions under which an object remains stationary relative to an accelerating surface.

Simple Harmonic Motion (SHM)

Simple Harmonic Motion describes periodic oscillatory motion where the restoring force is directly proportional to the displacement from equilibrium. In SHM, acceleration is proportional to displacement but opposite in direction, and key parameters such as period, amplitude, and angular frequency characterize the motion.

Angular Frequency and its Relationship to Period

Angular frequency, denoted by ?, is a measure of how rapidly oscillations occur in SHM and is related to the period T by the equation ? = 2?/T. This relationship is important because it allows us to connect the timing of the oscillation to the magnitude of acceleration experienced at the oscillation's extremum.

Maximum Acceleration in SHM

In SHM, the maximum acceleration occurs at the maximum displacement (amplitude) and is given by a_max = ?²A. Setting this maximum acceleration equal to the maximum allowable static frictional acceleration (?g) gives the condition necessary to prevent sliding. This conceptual understanding links the dynamics of SHM with the threshold of frictional resistance.

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