A body is placed on a horizontal plank executing S.H.M....

A body is placed on a horizontal plank executing S.H.M. along vertical direction. Its amplitude of oscillation is $3.92 \times 10^{-3} \mathrm{~m}$. What should be the minimum periodic time so that the body does not loose contact with the plank? (A) $0.1256 \mathrm{~s}$ (B) $0.1356 \mathrm{~s}$ (C) $0.1456 \mathrm{~s}$ (D) $0.1556 \mathrm{~s}$

A body is placed on a horizontal plank executing S.H.M. along vertical direction. Its amplitude of oscillation is $3.92 \times 10^{-3} \mathrm{~m}$. What should be the minimum periodic time so that the body does not loose contact with the plank?
(A) $0.1256 \mathrm{~s}$
(B) $0.1356 \mathrm{~s}$
(C) $0.1456 \mathrm{~s}$
(D) $0.1556 \mathrm{~s}$

Key Concepts

Simple Harmonic Motion

Simple Harmonic Motion (SHM) describes oscillatory motion where the restoring force is directly proportional to the displacement, leading to a sinusoidal motion in time. Understanding SHM is fundamental for analyzing systems where periodic behavior is observed, detailing how position, velocity, and acceleration vary over a cycle with a well-defined amplitude and period.

Maximum Acceleration in SHM

In SHM, the maximum acceleration occurs at the maximum displacement (amplitude) and is given by the formula a_max = A ?², where A is the amplitude and ? is the angular frequency. This concept is crucial in determining the forces experienced by an oscillating body, particularly in cases where the acceleration might exceed other influencing accelerations such as gravitational acceleration.

Period and Angular Frequency Relationship

The angular frequency (?) is directly related to the period (T) of oscillation by the expression ? = 2?/T. This relationship enables the conversion between the time domain (period) and the cyclical nature of the motion (angular frequency), allowing for the calculation of motion characteristics from one form of measurement to another.

Condition for Maintaining Contact in Oscillatory Motion

When analyzing systems where an object is supported by an oscillating platform, the condition for maintaining contact involves ensuring that the maximum acceleration of the platform does not exceed gravitational acceleration. Specifically, if the downward acceleration from the oscillation equals or surpasses the acceleration due to gravity, the object may lose contact. Equating the maximum acceleration (A ?²) to gravitational acceleration (g) provides the threshold condition required to maintain continuous contact during the oscillatory motion.

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