A block of mass m compresses a spring of stiffness k through...

A block of mass $m$ compresses a spring of stiffness $k$ through a distance $I / 2$ as shown in the figure. If the block is not fixed to the spring, the period of motion of the block is
(a) $2 \pi \sqrt{m / k}$
(b) $(\pi+4) \sqrt{m / k}$
(c) $(1+\pi) \sqrt{m / k}$
(d) None of these

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Key Concepts

Simple Harmonic Motion

Simple harmonic motion (SHM) describes the periodic, oscillatory motion where the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction. In SHM, the motion is sinusoidal and the period of the oscillation is independent of the amplitude, assuming small displacements.

Mass-Spring System

A mass-spring system is a classic example of a system exhibiting simple harmonic motion. In this system, a mass attached to a spring oscillates when displaced from its equilibrium position, with the spring force described by Hooke’s Law. The period of oscillation depends on the mass and the stiffness (spring constant) of the spring.

Oscillation Period

The oscillation period is the time required for one complete cycle of motion. For an ideal mass-spring system undergoing simple harmonic motion, the period is given by T = 2??(m/k), where m is the mass and k is the spring constant. This result is derived from the fundamental equation of motion for SHM.

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