Young's Modulus

"Young's modulus ($Y$) is the ratio of stress to strain for a particular material.  young's modulus is a property of the material, and does not depend on the size or shape of an object.  The stiffer the material, the larger is Young's modulus."

The strain on a material is the fractional stretch, or the change in length divided by the original length.  This gives a ratio defined as $$strain = \frac{\Delta L}{L}$$The stress on a material is the tension force per unit of area.  This eliminates the dependency of thickness of the material by using both the tension force ($F_T$) and cross-sectional area of a wire ($A$).$$stress = \frac{F_T}{A}$$These ratios can be related down to the atomic level, the stress ($\frac{F_T}{A}$) being the force that each chain of atomic bonds must exert, and the strain ($\frac{\Delta L}{L}$) being the stretch of the interatomic bond.  The stiffer the material, the larger the modulus.  We write young's modulus as$$\frac{F_T}{A} = Y\frac{\Delta L}{L}$$$$Y = \frac{stress}{strain} = \frac{\frac{F_T}{A}}{\frac{\Delta L}{L}}$$This is dependent upon the fact that too large a stress will result in the material coming apart and breaking, a process known as "yielding."  The Young's modulus in terms of atomic quantities can be defined as:$$Y = \frac{\frac{k_{s,i}s}{d^2}}{\frac{s}{d}} = \frac{k_{s,i}}{d}$$$d$ represents the relaxed length of an interatomic bond, and the diameter of one atom.  The cross-sectional area of an atom is $d^2$, considering we are viewing the atom as occupying a cube of space in the crystal lattice versus its approximately spherical shape.
The stretch of the interatomic bond is $s$, or the $\Delta L$.
The interatomic force is $k_{s,i}$.

Tension Forces

Tension Forces are just forces that compensate for other forces.  For instance, when a ball is hanging motionless on a wire, the net force on the ball must be zero.  The gravity of the Earth is pulling it down, therefore there must be another force in the opposite direction compensating for this.

The force exerted by an object such as a wire or string is called a tension.$$\Delta \vec{p} = \vec{F}_{net}\Delta t$$$$0 = (\vec{F}_T - \vec{F}_{ext})\Delta t$$

Conservation of Momentum

The concept of conservation of momentum is simple to understand by looking at the Momentum Principle.  The Momentum Principle predicts how much the momentum of a system will change based on external forces, or forces in the surroundings.  Momentum gained by the system is transferred from the surroundings, so we say the momentum is conserved.$$\Delta \vec{p}_{sys} = -\Delta \vec{p}_{surr}$$$$\Delta \vec{p}_{sys} + \Delta \vec{p}_{surr} = \vec{0}$$Basically, all force is equal and opposite.

Spring Force

The spring force was determined through experimentation to be accurately modeled by the equation:$$|\vec{F}_{spring}| = -k_s (|\vec{L}| - L_i)\hat{L}$$$$\vec{F} = -k_s s \hat{L}$$It's easy to determine from this equation:$$s = |\vec{L}| - L_i$$
$k_s$ is the spring constant, or stiffness, of the spring.  It dictates the stretching capabilities of the spring.
$L_i$ is the length of the relaxed spring (no compression/extension).
$\vec{L}$ extends from the point of attachment of the spring to the mass at the other end (the total, final length).
The stretch of $s$ can be negative or positive (compression or extension).

Impulse

Impulse is the product of the force ($\vec{F}$) and change in time ($\Delta t$).  $$Impulse = \vec{F}\Delta t$$The impulse differs from the Momentum Principle in that it is distinct in forces.  The change in momentum ($\Delta\vec{p}$) is equal to the sum of the impulses applied, or:  $$\Delta\vec{p} = \sum\limits_{i=1}^n \vec{F}_i\Delta t$$

Force

The concept of force was created to quantify the interaction between 2 objects.  Force is regarded as a vector because it has a magnitude and is exerted in a specific direction.  Force is most easily measured by utilizing a spring - when we stand on a spring scale, it is merely measuring the force of our weight due to gravity pressing against it.  By measuring how much a spring compresses when a force is applied we can obtain a very accurate measurement - an equation found (link to post) here.

Force is largely measured in newtons (N).  1 N is approximately the downward gravitational force of the Earth on a small apple (Newton formulated the idea of gravity by watching an apple fall from a tree, what a coincidence).

Force is the derivative of momentum, described as "the instantaneous time rate of change of the momentum of an object is equal to the net force acting on the object," or more simply, "the derivative of the momentum with respect to time is equal to the net force acting on the object."$$\frac{d\vec{p}}{dt} = \vec{F}_{net}$$

The net force ($\vec{F}_{net}$) is widely used in physics to describe all the forces acting on a system:  $$\vec{F}_{net} =  \sum\limits_{i=1}^n \vec{F}_i$$