A 2.5-kg body undergoes SHM and makes exactly 3 vibrations...

A $2.5-\mathrm{kg}$ body undergoes SHM and makes exactly 3 vibrations each second. Compute the acceleration and the restoring force acting on the body when its displacement from the equilibrium position is 5.0 cm .

Key Concepts

Simple Harmonic Motion (SHM)

SHM is a type of periodic motion where the restoring force is directly proportional to the displacement from the equilibrium position, but opposite in direction. This concept is critical as it forms the basis for analyzing the body’s oscillatory motion, energy transfer, and the forces involved.

Angular Frequency

Angular frequency, denoted by ?, represents how fast the oscillation occurs in radians per second. It is related to the frequency by the formula ? = 2?f. This concept is essential for converting the number of vibrations per second into a parameter that can be used in calculations involving acceleration and restoring force.

Acceleration in SHM

In SHM, the acceleration is proportional to the displacement from the equilibrium position but acts in the opposite direction. This relationship is mathematically expressed as a = -?²x. Understanding this concept allows one to calculate the acceleration at any given displacement, highlighting the intrinsic link between displacement and acceleration in oscillatory systems.

Restoring Force

The restoring force in SHM is the force that brings the object back toward its equilibrium position. It is directly related to the acceleration by Newton’s second law, F = ma, and is often expressed as F = -m?²x when using the SHM framework. This concept is crucial for understanding how the properties of the system, such as mass and angular frequency, influence the magnitude of the force acting to restore equilibrium.

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