A rigid block of mass M is mounted on four elastic supports,...

A rigid block of mass $M$ is mounted on four elastic supports, as shown in Fig. $2.58 .$ A mass $m$ drops from a height $l$ and adheres to the rigid block without rebounding. If the spring constant of each elastic support is $k$, find the natural frequency of vibration of the system (a) without the mass $m,$ and (b) with the mass $m$. Also find the resulting motion of the system in case (b).

Key Concepts

Simple Harmonic Motion

Simple harmonic motion (SHM) describes the periodic oscillatory movement of systems where the restoring force is directly proportional to the displacement from equilibrium. The natural frequency of such a system is determined by the square root of the ratio of the effective spring constant to the mass involved, leading to oscillations that can be described by sine and cosine functions.

Equivalent Spring Constant

When multiple elastic supports or springs work together, their individual stiffnesses combine to form an effective spring constant. In a parallel configuration, the effective stiffness is the sum of the individual spring constants. This combined stiffness plays a crucial role in determining the oscillation frequency of the overall system.

Mass-Spring System Dynamics

A mass-spring system is a foundational model in mechanics where a mass attached to a spring exhibits oscillatory motion when displaced from equilibrium. The dynamics of the system are governed by Newton's second law, resulting in a differential equation whose solution gives the system’s natural frequency and amplitude, characterizing its vibratory behavior.

Inelastic Collision and Effective Mass Change

An inelastic collision occurs when colliding bodies stick together, causing a change in the effective mass of the system that subsequently vibrates. Although energy is not conserved during the collision, momentum conservation helps determine the new amplitude and phase of the oscillatory motion, as well as altering the natural frequency due to the increased mass.

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