Chapter 9

Static Equilibrium; Elasticity and Fracture

Equilibrium

• First Condition of Equilibrium– The net external force must be zero

• Second Condition of Equilibrium– The net external torque must be zero

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Torque and Equilibrium, cont

• To ensure mechanical equilibrium, you need to ensure rotational equilibrium as well as translational

• The Second Condition of Equilibrium states– The net external torque must be zero

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9-1 The Conditions for Equilibrium

The first condition for equilibrium is that the forces along each coordinate axis add to zero.

9-2 Solving Statics ProblemsThe previous technique may not fully solve all statics problems, but it is a good starting point.

9.2 Rigid Objects in Equilibrium

Example 3 A Diving Board

A woman whose weight is 530 N is poised at the right end of a diving boardwith length 3.90 m. The board hasnegligible weight and is supported bya fulcrum 1.40 m away from the leftend.

Find the forces that the bolt and the fulcrum exert on the board.

9.2 Rigid Objects in Equilibrium

022 WWF

N 1480m 1.40

m 90.3N 5302 F

22

WWF

9.2 Rigid Objects in Equilibrium

021 WFFFy

0N 530N 14801 F

N 9501 F

9-2 Solving Statics ProblemsIf there is a cable or cord in the problem, it can support forces only along its length. Forces perpendicular to that would cause it to bend.

9-1 The Conditions for EquilibriumThe second condition of equilibrium is that there be no torque around any axis; the choice of axis is arbitrary.

9-2 Solving Statics Problems

1. Choose one object at a time, and make a free-body diagram showing all the forces on it and where they act.

2. Choose a coordinate system and resolve forces into components.

3. Write equilibrium equations for the forces.

4. Choose any axis perpendicular to the plane of the forces and write the torque equilibrium equation. A clever choice here can simplify the problem enormously.

5. Solve.

9-2 Solving Statics ProblemsIf a force in your solution comes out negative (as FA will here), it just means that it’s in the opposite direction from the one you chose. This is trivial to fix, so don’t worry about getting all the signs of the forces right before you start solving.

9-3 Applications to Muscles and Joints

These same principles can be used to understand forces within the body.

9-3 Applications to Muscles and Joints

The angle at which this man’s back is bent places an enormous force on the disks at the base of his spine, as the lever arm for FM is so small.

9-4 Stability and BalanceIf the forces on an object are such that they tend to return it to its equilibrium position, it is said to be in stable equilibrium.

9-4 Stability and Balance

If, however, the forces tend to move it away from its equilibrium point, it is said to be in unstable equilibrium.

9-4 Stability and Balance

An object in stable equilibrium may become unstable if it is tipped so that its center of gravity is outside the pivot point. Of course, it will be stable again once it lands!

9-4 Stability and BalancePeople carrying heavy loads automatically adjust their posture so their center of mass is over their feet. This can lead to injury if the contortion is too great.

9-5 Elasticity; Stress and Strain

Hooke’s law: the change in length is proportional to the applied force.

(9-3)

9-5 Elasticity; Stress and StrainThis proportionality holds until the force reaches the proportional limit. Beyond that, the object will still return to its original shape up to the elastic limit. Beyond the elastic limit, the material is permanently deformed, and it breaks

at the breaking point.

9-5 Elasticity; Stress and StrainThe change in length of a stretched object depends not only on the applied force, but also on its length and cross-sectional area, and the material from which it is made.

The material factor is called Young’s modulus, and it has been measured for many materials.

The Young’s modulus is then the stress divided by the strain.

9-5 Elasticity; Stress and Strain

In tensile stress, forces tend to stretch the object.

9-5 Elasticity; Stress and Strain

Compressional stress is exactly the opposite of tensional stress. These columns are under compression.

9-5 Elasticity; Stress and Strain

Shear stress tends to deform an object:

9-6 Fracture

If the stress is too great, the object will fracture. The ultimate strengths of materials under tensile stress, compressional stress, and shear stress have been measured.

When designing a structure, it is a good idea to keep anticipated stresses less than 1/3 to 1/10 of the ultimate strength.

9-6 Fracture

A horizontal beam will be under both tensile and compressive stress due to its own weight.

9-6 Fracture

What went wrong here? These are the remains of an elevated walkway in a Kansas City hotel that collapsed on a crowded evening, killing more than 100 people.

9-6 Fracture

Here is the original design of the walkway. The central supports were to be 14 meters long.

During installation, it was decided that the long supports were too difficult to install; the walkways were installed this way instead.

9-1 The Conditions for EquilibriumAn object with forces acting on it, but that is not moving, is said to be in equilibrium.

9-6 Fracture

The change does not appear major until you look at the forces on the bolts:

The net force on the pin in the original design is mg, upwards.

When modified, the net force on both pins together is still mg, but the top pin has a force of 2mg on it – enough to cause it to fail.

9-7 Spanning a Space: Arches and Domes

The Romans developed the semicircular arch about 2000 years ago. This allowed wider spans to be built than could be done with stone or brick slabs.

9-7 Spanning a Space: Arches and DomesThe stones or bricks in a round arch are mainly under compression, which tends to strengthen the structure.

9-7 Spanning a Space: Arches and Domes

Unfortunately, the horizontal forces required for a semicircular arch can become quite large – this is why many Gothic cathedrals have “flying buttresses” to keep them from collapsing.

9-7 Spanning a Space: Arches and Domes

Pointed arches can be built that require considerably less horizontal force.

9-7 Spanning a Space: Arches and Domes

A dome is similar to an arch, but spans a two-dimensional space.

Summary of Chapter 9

• An object at rest is in equilibrium; the study of such objects is called statics.

• In order for an object to be in equilibrium, there must be no net force on it along any coordinate, and there must be no net torque around any axis.

• An object in static equilibrium can be either in stable, unstable, or neutral equilibrium.

Summary of Chapter 9

• Materials can be under compression, tension, or shear stress.

• If the force is too great, the material will exceed its elastic limit; if the force continues to increase the material will fracture.