Chapter 12: Equilibrium and Elasticity



Conditions Under Which a Rigid Object is in Equilibrium



Problem-Solving Strategy



Elasticity

Equilibrium:

An object at equilibrium is either ...

• at rest and staying at rest (i.e., static equilibrium) , or • in motion and continuing in motion with the constant velocity and constant angular momentum.

constant.

P



L

 

m r



v

  

P



m

(

r

 

v

 center of mass, or any other point, is constant.

)

Conditions of Equilibrium:

 Net force: Net torque:

net



d P dt m



a

 

net

 

d L dt



r

  

m a

Conditions of equilibrium: 

F net

 0 (balance of forces) and   net

F net

,

x

 0 

net

,

x



F net

,

F net

,

z y

  0 0  

net

,

net

,

z y

0 (balance    0 0 0 of torques) Another requirements for static equilibrium: 

P

 0

The center or gravity:

The gravitational force on a body effectively acts at a single point, called the

center of gravity

(cog) of the body.

•the

center of mass

of an object depends on its shape and its density •the

center of gravity

of an object depends on its shape, density, and the external gravitational field. Does the center of gravity of the body always coincide with the

center of mass

(com)? Yes, if the body is in a uniform gravitational field.

How is the center of gravity of an object determined?

The

center of gravity

(

cog)

of a regularly shaped body of uniform composition lies at its geometric center.

The (cog) of the body can be located by suspending it from several different points. The

cog

is always on the line-of action of the force supporting the object.

cog

Problem-Solving Strategy:

• Define the system to be analyzed • Identify the forces acting on the system • Draw a free-body diagram of the system and show all the forces acting on the system, labeling them and making sure that their points of application and lines of action are correctly shown.

• Write down two equilibrium requirements in components and solve these for the unknowns

Sample Problem 12-1:

• Define the system to be analyzed: beam & block • Identify the forces acting on the system: the gravitational forces: mg & Mg, the forces from the left and the right scales: F l & F r • Draw a force diagram • Write down the equilibrium requirements in components and solve these for the unknowns

O

balance of forces balance of torques :

F l

 :

LF r F r

  ( 1 2

Mg L

) 

mg mg

 ( 1 4  0

L

)

Mg

 0

F l

 0

Elasticity

Some concepts : • Rigid Body: • Deformable Body: elastic body: rubber, steel, rock… plastic body: lead, moist clay, putty… • Stress: Deforming force per unit area (N/m 2 ) • Strain: unit deformation

Elastic modulus



Stress Strain

Young’s Modulus: Elasticity in Length

The Young’s modulus, E, can be calculated by dividing the stress by the strain, i.e.

E



stress strain



F

/ 

L

/

A L



F A L



L

where (in SI units)

E

is measured in newtons per square metre (N/m²).

F

is the force, measured in newtons (N)

A

is the cross-sectional area through which the force is applied, measured in square metres (m 2 ) 

L

is the extension, measured in metres (m)

L

is the natural length, measured in metres (m)

Table 12-1: Some elastic properties of selected material of engineering interest Material Steel Aluminum Glass Concrete Wood Bone Polystyrene Density  (kg/m 3 ) 7860 Young’s Modulus E (10 9 N/m 2 ) 200 Ultimate Strength S u (10 6 N/m 2 ) 400 Yield Strength S y (10 6 N/m 2 ) 250 2710 2190 2320 525 1900 1050 70 65 30 13 9 3 110 50 40 50 170 48 90     

Shear Modulus: Elasticity in Shape

The shear modulus, G, can be calculated by dividing the shear stress by the strain, i.e.

G

 shear stress shear strain 

F

/ 

x

/

A L



F A L



x

where (in SI units)

G

is measured in newtons per square metre (N/m²)

F

is the force, measured in newtons (N)

A

is the cross-sectional area through which the force is applied, measured in square metres (m 2 ) 

x

is the horizontal distance the sheared face moves, measured in metres (m)

L

is the height of the object, measured in metres (m)

Bulk Modulus: Elasticity in Volume

The bulk modulus, B, can be calculated by dividing the hydraulic stress by the strain, i.e.

B

 hydraulic pressure hydraulic strain  

V p

/

V



p V



V

where (in SI units)

B

is measured in newtons per square metre (N/m²)

P

is measured in in newtons per square metre (N/m²) 

V

is the change in volume, measured in metres (m 3 )

V

is the original volume, measured in metres (m 3 )

Young’s modulus

Under tension and compression

Shear modulus

Under shearing

Bulk modulus

Under hydraulic stress Strain is 

L

/

L E



Stress Strain



F A L



L

Strain is 

x

/

L G



Stress Strain



F A L



x

Strain is 

V

/

V B



Stress Strain



p V



V

Summary:

• Requirements for Equilibrium: 

F net

 0 and   net  0 • The

cog

of an object coincides with the

com

if the object is in a uniform gravitational field.

• Solutions of Problems: •Define the system to be analyzed • Identify the forces acting on the system • Draw a force diagram • Write down the equilibrium requirements in components and solve these for the unknowns • Elastic Moduli: tension and compression shearing hydraulic stress

p



B



V V

stress

F A



G



x L F



A

 modulus 

L E L

 strain

Sample Problem 12-2:

• Define the object to be analyzed: firefighter & ladder • Identify the forces acting on the system: the gravitational forces: mg & Mg, the force from the wall: F w the force from the pavement: F px & F py • Draw a force diagram • Write down the equilibrium requirements in components and solve these for the unknowns balance of forces :

F w



F px

 0

F py

balance of torques 

Mg



mg

 0 : ( 1 2

a

)

Mg

 ( 1 3

a

)

mg



hF w

 0 where,

a



L

2 

h

2

Sample Problem 12-3:

• Define the object to be analyzed: Beam • Identify the forces acting on the system: the gravitational force (mg), the force from the rope (T r ) the force from the cable (T c ), and the force from the hinge (F v and F h ) • Draw a force diagram

• Write down the equilibrium requirements in components and solve these for the unknowns Balance of torques: 

net

,

z



aT c



bT



r

T c  ( 1 2

b

)(

mg

) 

gb

(

M a

 1 2  0

m

)  6093

N

Balance of forces:

F net

,

x



F h



T



c



F h

0 

T c

 6093

N F net

,

y



F v



mg



T r



F v

 0 

mg



Mg

 5047

N F



F h

2 

F v

2  7900

N

Sample Problem 12-6:

• Define the system to be analyzed: table plus steel cylinder.

• Identify the forces acting on the object: the gravitational force (Mg), the forces on legs from the floor (F 1 = F 2 = F 3 and F 4 ).

• Draw a force diagram 

F

1 3

F

3  

F

2

F

4 

F

3 

F

4

F g



Mg

• Write down the equilibrium requirements in components and solve these for the unknowns Balance of forces:

F net

 3

F

3 

F

4 

Mg

 0 If table remains level:       

F A F

4

A L

3 4   

E



L



L L E



L

3

L

 4 3

d



F

4

L AE



F

3

L



d AE F F

4 3  550

N

 1200

N