A 25-kg block is supported by the spring arrangement shown....

A $25-\mathrm{kg}$ block is supported by the spring arrangement shown. If the block is moved vertically downward from its equilibrium position and released, determine ( $a$ ) the period and frequency of the resulting motion, $(b)$ the maximum velocity and acceleration of the block if the amplitude of the motion is $30 \mathrm{mm} .$

Key Concepts

Simple Harmonic Motion (SHM)

SHM describes the behavior of a system where the restoring force is directly proportional to the displacement from its equilibrium position and acts in the opposite direction. This concept underpins the analysis of oscillatory systems like mass-spring setups, where the motion can be described by sinusoidal functions.

Angular Frequency

Angular frequency, denoted by ?, quantifies how fast the system oscillates and is defined in a mass-spring system as ? = ?(k/m), where k is the spring constant and m is the mass. It serves as the cornerstone for calculating other dynamic parameters in SHM, such as period and maximum speed.

Period and Frequency

The period (T) is the time it takes for the oscillating system to complete one full cycle, with T = 2?/?. The frequency (f), defined as the number of complete cycles per unit time, is the reciprocal of the period (f = 1/T). These concepts are fundamental in characterizing the temporal behavior of oscillatory systems.

Amplitude

Amplitude refers to the maximum displacement from the equilibrium position in an oscillatory system. It is a critical parameter because it influences the maximum velocity and acceleration achieved during the motion, even though it does not affect the period in ideal simple harmonic motion.

Maximum Velocity and Acceleration in SHM

In SHM, the maximum velocity occurs as the object passes through the equilibrium position and is given by v_max = ?A, where A is the amplitude. Conversely, the maximum acceleration occurs at the points of maximum displacement and is computed as a_max = ?²A. These expressions are vital for understanding the dynamics of oscillating systems.

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