Conservation of Energy in SHM

 Conservation of energy applies to simple harmonic motion.
An object held at its highest point (maximum displacement) will have reached its maximum potential energy, and its kinetic energy will be zero due to having no velocity.
As the object moves towards its equilibrium position and accelerates, its kinetic energy steadily rises due to increasing velocity, and its potential energy decreases as the object descends and loses height.

The maximum kinetic energy is achieved at the object's equilibrium position where it has the highest velocity.
This maximum kinetic energy is equal to the object's maximum potential energy at its highest point.
In these scenarios of SHM, air resistance and any other external force is considered negligible, therefore the total energy in the system always remains the same.

Simple Harmonic Motion

Simple harmonic motion is a motion of constant amplitude for which the acceleration is proportional but oppositely directed to the displacement from the equilibrium position (meaning that even though a and s are proportional, one will be positive and one will be negative).

An object continuously oscillates from a maximum point to an equilibrium position to a minimum point and back. This movement can be modeled as y=sinx or y=cosx.

s/t, v/t and a/t graphs can be created for simple harmonic motion.

The s/t graph can be differentiated to show the v/t graph, and the v/t graph can be differentiated to show the a/t graph. Integration can be used to go back from a/t to v/t and from v/t to s/t graphs.

Momentum

Momentum is equal to mass of an object multiplied by its velocity. It is a vector quantity.

P=mv

The principle of conservation of momentum states that the momentum of a system before a collision is equal to the momentum of a system after the collision. In other words:
m1u1 + m2u2 = m1v1 + m2v2

In other circumstances, the colliding objects may coalesce (join together, for example a train and a carriage). 
Conservation of momentum still applies, but in this manner:
m1u1 + m2u2 = (m1 + m2)v

There are two types of collisions:

• Elastic collisions are collisions in which momentum as well as energy are conserved.
• Inelastic collisions are collisions in which momentum is also conserved, but energy is dissipated in the form of heat, sound, light, friction, etc.

Circular Motion in everyday life

Circular motion is seen in many things in everyday life, such as washing machines.

Washing machines can dry clothes by using circular motion.

The clothes inside the machine have a centripetal force pulling them towards the centre of the tub, causing them to continuously spin.

However, the water does not have a centripetal force, therefore the water leaves the path of circular motion via the holes inside the tub.

The water leaves the tub along its path of velocity, which is always tangential to the radius (as shown on the diagram).