Motion of a particle on a light inextensible string

Motion of a Particle on a Light, Inextensible String

A light, inextensible string means that the string has no mass and does not stretch. Any motion of the particle will not be affected by the characteristics of the string beyond its length.
When a particle is suspended from this string and subjected to force, it can exhibit different types of motion, typically simple harmonic or circular motion.

The weight of the particle, acting downwards, is represented as mg, where m is the mass of the particle and g is the gravitational acceleration.
Tension in the string acts along the string. Its direction will depend on the current position of the particle. In equilibrium conditions, the tension is equal and opposite to the weight.

If the particle is made to move in a circle (horizontally or vertically), it executes a circular motion.
The tension in the string provides the centripetal force needed for circular motion.
The equation of motion here becomes T - mg cos(θ) = mω²r, where T is the tension in the string, θ is the angle between the string and the vertical, ω is the angular speed, and r is the radius of the circular path.

If a particle is momentarily displaced and left to move under gravity, it will execute a simple harmonic motion.
The forces involved in simple harmonic motion are the weight of the particle and the tension in the string.
Analysing the forces vertically and horizontally can help derive the equation of motion.
For small displacements, the motion is well approximated by: a = -g sin(θ), where a is acceleration, g is the gravitational acceleration, and θ is the displacement angle. This equation describes the simple harmonic motion.

Resolving forces is the process of breaking a force into vertical and horizontal components for ease of analysis.
When a particle on a string is displaced, the tension in the string can be resolved into the components: T cos(θ) (vertical) and T sin(θ) (horizontal).

The kinetic energy of the particle increases when the potential energy decreases and vice versa. This is due to the conservation of mechanical energy, assuming the system is frictionless.
The total energy of the particle (kinetic energy + potential energy) remains constant. This can be useful for solving problems particularly when the particle is at the two extreme points (highest speed or maximum displacement).