Chapter 12

Equilibrium and Elasticity

12.1 Rigid Object in Equilibrium

12.3Examples of Rigid Objects in Static Equilibrium

12.4 Elastic Properties of Solids

Conditions for equilibrium

An example of is shown in the figures.

In fig. a we balance a domino with the domino's center of mass

vertically above the supporting edge. The torque of the

g

un

rav

stable equil

itational fo

ibrium

rce

about the supporting edge is zero

because the line of action of passes through the edge.

g

g

F

F

Examples of Rigid Objects in Static Equilibrium

Statics Problem Recipe

1. Draw a force diagram. (Label the axes.) 2. Choose a convenient origin O. A good choice is to have one of the unknown forces acting at O. 3. Sign of the torque for each force:

- If the force induces clockwise (CW) rotation + If the force induces counterclockwise (CCW) rotation

4. Equilibrium conditions:

5. Make sure that number of unknowns = number of equations

net , net,

net ,

0 0

0

x y

z

F F

The gravitational force acting on an extended body is the vector sum of the

gravitational forces acting on the individual elements of the body. The gravitational

force gF

The Center of Gravity (cog)

on a body effectively acts at a single point known as the center of gravity

of the body. Here "effect If the individual

gravitational

ively" has the following meaning

forces on the elements of th

:

e body are turned off and replaced by

acting at the center of gravity, then the net force and the net torque about any point

on the body do not change. We shall prove that if the acceleration of g av r i

gF

center of gravity

cente

ty is

the same for all the elements of the body then the coincides with

the . This is a reasonable approximation for objects near the surface

of the Earth

r o

be

f mass

cause

g

changes very little.g

Elastic Properties of Solids

STRESS• Stress is the force that produces strain on a physical body

without undergoing some sort of physical change.

Stress = σ = F/A• UNIT: N/m2 or Pascal

STRAINStrain is the deformation of a physical body under the action of applied forces

The tendency of a body to return to its original shape after it has been stretched or compressed

Elasticity

The physical property of being stiff and resisting bending

Rigidity

Stress and strain are directly proportional to each other. The constant of proportionality is called a modulus of elasticity, so that

Stress = Elastic modulus × Strain

We consider three types of deformation and define an elastic modulus for each:

1. Young’s modulus2. Shear modulus3. Bulk modulus

Young’s modulus or Elasticity in LengthIt measures the resistance of a solid to a change in its length.

Shear modulus or Elasticity of Shape It measures the resistance to motion of the planes within a solid parallel to each other.

Bulk modulus or Volume Elasticity It measures the resistance of solids or liquids to changes in their volume.