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EXTENSIVE stretching doesn’t obey (F/A)= Y (∆l/l)….elastic limit: after this strain threshold which the equation is no longer applicable!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

Fext= 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)

Felast= -k(x-xeq) <- (RESTORING FORCE)

Cases:

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

2) When position x<xeq the Felast 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

Eelast= 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.

Etotal= CONSTANT= Ekin + Eelastic

Eelastic= 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

Etotal= (1/2)k x A^2…………………………A= [(2 x Etotal)/k]^1/2

HENCE: vmax= (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

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