**Stress: **force exerted per unit area of surface on an extended object. **STRESS LEAD TO STRAIN **

Three types of deformation due to strain:

- Stretching (change in length)
- Twisting (change in angle)
- Compression or Expansion (change in volume)

****Each of these strains is caused by a different form of stress *

**Tensile stress=**Stretching**Shearing stress=**Twisting**Hydraulic stress=**Expansion or Compression

## Tensile Stress and Stretching

**Stretching: **stressing in vertical direction, caused by two opposite force, **F1 **& **F2**, DENSITY is **Constant**(this only works for solids and liquids)

Strain is the relative change in length, express by ∆l/l, there strain= **DIMENSIONLESS QUANTITY **

**Tensile Stress**= F/A **F1=-F2- **units for pressure (Pa) hence related by Newton’s **first law **but not Newton’s **third law…**forces act on the same object…

**Elastic: **strain and stress are proportional of each other (i.e when length change of object (strain) increases linearly with the stress)

(F/A)= Y (∆l/l) [**Ơ= Y x Ɛ**]

**Y=**Young’s modulus (proportionality factor)**units:**Pa**Ơ=**Stress (F/A)**Ɛ=**∆l/l

**Large **Young’s moduli= **STRONG MATERIAL! **Even large force acting on a piece of material leads to a small length increase

**Small **Young’s moduli= **SOFT MATERIAL! **Even small force acting on piece of material leads to large length increase

**EXTENSIVE **stretching doesn’t obey (F/A)= Y (∆l/l)….**elastic limit: **after this strain threshold which the equation is no longer applicable!

**LARGE **strain below strain threshold…..*permanent plastic deformations *

**PLASTIC DEFORMATIONS**

**Permanent Plastic deformation- **once you pass the elastic point of a material and thought is **ultimate strength point **

## Hooke’s Law

Applies in the elastic regime of a material and states that the stress of the material are linearly proportional.

**F**ext= **k**(x-xeq)

**k= **spring constant in units N/m (*the Fext is linearly proportional to the displacement…takes twice as much force to stretch a spring twice as far) *

**F**elast= **-k**(x-xeq) <- (RESTORING FORCE)

**Cases: **

**1) **When position **x> xeq** the **F**elast is **negative**, force pulls the object to back to a **smaller** value

**2) **When position **x<xeq** the **F**elast is **positive**, the force pushes the object toward a **larger **value

*This works for ideal spring because REAL springs don’t follow this law.*

**OBJECT ATTACHED TO AN IDEAL SPRING**

**Elastic (potential) energy: **dependent on object’s relative position to its equilibrium position

**E**elast= ½ k (x-xeq)^2

*A system with a linear restoring force (Hooke’s law) has elastic potential energy that is proportional to the square of the displacement from the equilibrium position of the system.*

**E**total= CONSTANT= **E**kin + **E**elastic

**E**elastic= HIGHEST at release point & collapse (two points)

**Amplitude (A): **the maximum displacement of a vibration object (*when object has moved to the amplitude, its kinetic energy becomes ZERO, Ekin=0 and the Etotal= Eelastic*

**E**total= (1/2)k x A^2…………………………A= [(2 x **E**total)/k]^1/2

HENCE: **v**max= (k/m)^1/2 x A

**Simple Harmonic Motion: **an object oscillates about a point of mechanical equilibrium

**Frequency (angular): **Ѡ= (k/m)^1/2 **unit: **1/s

**Frequency: **(Hertz) f= 1/T