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Het basisvak Toegepaste Natuurwetenschappen http://www.phys.tue.nl/nfcmr/natuur/collegenatuur.html Applied Natural Sciences Leo Pel e mail: 3nab0@tue.nl http://tiny.cc/3nab0
Chapter 11 Equilibrium and Elasticity PowerPoint Lectures for University Physics, Thirteenth Edition Hugh D. Young and Roger A. Freedman Lectures by Wayne Anderson Copyright 2012 Pearson Education Inc.
LEARNING GOALS The conditions that must be satisfied for a body or structure to be in equilibrium. What is meant by the center of gravity of a body, and how it relates to the body s stability. How to solve problems that involve rigid bodies in equilibrium. How to analyze situations in which a body is deformed by tension, compression, pressure, or shear. What happens when a body is stretched so much that it deforms or break.
Introduction Many bodies, such as bridges, aqueducts, and ladders, are designed so they do not accelerate. Real materials are not truly rigid. They are elastic and do deform to some extent. We shall introduce concepts such as stress and strain to understand the deformation of real bodies.
Balancing act 6
Conditions for equilibrium First condition: The sum of all the forces is equal to zero: F x = 0 F y = 0 F z = 0 Second condition: The sum of all torques about any given point is equal to zero.
Two Conditions for Equilibrium F ext 0 ext 0 (about anypoint! ) When applying these, we must consider all external forces But the gravitational force is rather subtle
Center of gravity We can treat a body s weight as though it all acts at a single point the center of gravity. If we can ignore the variation of gravity with altitude, the center of gravity is the same as the center of mass.
Exception CM very tall buildings g uniform over body center of gravity =center of mass Petronas tower center of gravity = 2 cm under center of mass.
Center of mass: stable equilibrium We consider the torque created by the gravity force (applied to the CM) and its direction relative to the possible point(s) of rotation
Center of mass: stable equilibrium We consider the torque created by the gravity force (applied to the CM) and its direction relative to the possible point(s) of rotation
Center of mass: stable equilibrium We consider the torque created by the gravity force (applied to the CM) and its direction relative to the possible point(s) of rotation
Using the Center of Gravity
Example? max
Example X=0 Problem if x cg =x s
Example
Example
Strain, stress, and elastic moduli Stretching, squeezing, and twisting a real body causes it to deform. We shall study the relationship between forces and the deformations they cause. Stress is the force per unit area and strain is the fractional deformation due to the stress. Elastic modulus is stress divided by strain. The proportionality of stress and strain is called Hooke s law.
Stress and Strain When force is removed, the object will usually return to its original shape and size Matter is elastic We characterise the elastic properties of solids in terms of stress (=amount of force applied) and strain (=extent of deformation) that occur. The amount of stress required to produce a particular amount of strain is a constant for a particular material: this constant is called the elastic modulus Elastic Modulus stress strain
Elastic behaviour
Young s modulus Thomas Young (1773 1829) N/m 2 = Pascal (Pa)
Tensile and compressive stress and strain Tensile stress = F /A and tensile strain = l/l 0. Compressive stress and compressive strain are defined in a similar way. Young s modulus is tensile stress divided by tensile strain, and is given by Y = (F /A)(l 0 / l).
Some values of elastic moduli
Tensile stress and strain In many cases, a body can experience both tensile and compressive stress at the same time
Pressure pressure: N/m 2 = Pa 1 atmosfeer = 1 atm = 1.013 x 10 5 Pa
Bulk modulus Bulk modulus Volume change (relative) k 1 B compressibility: 1/Pa
Shear stress and strain 28
Sheer stress and strain Shear stress is F /A and sheer strain is x/h, Shear modulus is sheer stress divided by sheer strain, and is given by
Elastische moduli Rubber (PDMS) 0.3-10x10 6 0.25x10 10 0.1-3x10 6
Young s modulus Shear modulus Bulk modulus Under tension and compression Under shearing Under hydraulic stress L / L x / Strain is Strain is Strain is L V / V E Stress Strain F A L L G Stress Strain F A L x B Stress Strain p V V
Elasticity and plasticity Hooke s Law and Beyond O to b : elastic, reversible small stress, strain Hooke s Law: stress=modulus strain b to d: plastic, irreversible
Plastic deformation
Elasticity and plasticity
Summary