Solve problems involving osciliation of a particle attached to the end of an elastic spring or string

Understanding the Oscillation of a Particle Attached to a Spring or String

A particle attached to an elastic spring or string exhibits Simple Harmonic Motion (SHM) when displaced from its equilibrium position.
This system’s equilibrium position is the point where the spring or string is at rest and not stretched or compressed.
Displacement is the distance from the equilibrium position to the particle’s current position.
The particle experiences a restoring force proportional to its displacement and directed towards the equilibrium position. This force obeys Hooke’s Law given by F = -kx where k is the spring constant and x is the displacement.

The velocity of the particle is highest at the equilibrium position and lowest (zero) at the amplitude (maximum displacement).
The acceleration of the particle is highest at the amplitude (directed towards the equilibrium position) and zero at the equilibrium.
The period of the oscillation is given by T=2π sqrt(m/k) where m is the mass of the particle and k is the spring constant.
The frequency of the oscillations is the reciprocal of the period. It’s given by f=1/T.

Start with a diagram to visualize the problem and understand the forces involved.
Determine the initial conditions of the particle – its initial displacement, initial velocity, and overall displacement from equilibrium.
Use Hooke’s Law F = -kx to find the spring constant k if you’re given the restoring force and displacement.
Identify key parameters: mass of the particle m, spring constant k, and any given time intervals.
Use the formula for the period of oscillation T=2π sqrt(m/k) to find the unknown if two quantities are given.
Note that the period is independent of the amplitude.
Use energy conservation principles to solve problems involving height, speed, potential energy, and kinetic energy at different points of the oscillation.
Remember to apply the correct units for each quantity and carefully manage them during calculations.

In real-world scenarios, the system will experience damping due to external forces that dissipate energy, such as air resistance or internal friction within the spring.
Damping causes a gradual decrease in the amplitude of the oscillations, slowing down the motion.
Depending on the nature of damping, the system may approach its equilibrium position in different manners – under-damped, over-damped, or critically damped.
Understanding the concept of damping is critical for analysing and predicting the system’s behaviour over time.