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Introduction to Static Equilibrium and Elasticity | PHYS 130, Study notes of Physics

Material Type: Notes; Professor: Finn; Class: General Physics I; Subject: Physics; University: Siena College; Term: Unknown 1989;

Typology: Study notes

Pre 2010

Uploaded on 08/09/2009

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Chapter 12
Static Equilibrium
and
Elasticity
Static Equilibrium
Equilibrium implies the object is at rest (static)
or its center of mass moves with a constant
velocity (dynamic)
Static equilibrium is a common situation in
engineering
Principles involved are of particular interest to
civil engineers, architects, and mechanical
engineers
Equilibrium Summary
There are two necessary conditions for equilibrium
The resultant external force must equal zero:
ΣF = 0
This is a statement of translational equilibrium
The acceleration of the center of mass of the object must be
zero when viewed from an inertial frame of reference
The resultant external torque about any axis must
be zero:
Στ = 0
This is a statement of rotational equilibrium
The angular acceleration must equal zero
Static vs. Dynamic Equilibrium
In this chapter, we will concentrate on static
equilibrium
The object will not be moving
Dynamic equilibrium is also possible
The object would be rotating with a constant
angular velocity
In either case, the Στ = 0
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Chapter 12

Static Equilibrium and Elasticity

Static Equilibrium

  • Equilibrium implies the object is at rest (static) or its center of mass moves with a constant velocity (dynamic)
  • Static equilibrium is a common situation in engineering
  • Principles involved are of particular interest to civil engineers, architects, and mechanical engineers

Equilibrium Summary

  • There are two necessary conditions for equilibrium
  • The resultant external force must equal zero: Σ F = 0
  • This is a statement of translational equilibrium
  • The acceleration of the center of mass of the object must be zero when viewed from an inertial frame of reference
  • The resultant external torque about any axis must be zero: Στ = 0
  • This is a statement of rotational equilibrium
  • The angular acceleration must equal zero

Static vs. Dynamic Equilibrium

  • In this chapter, we will concentrate on static equilibrium - The object will not be moving
  • Dynamic equilibrium is also possible
    • The object would be rotating with a constant angular velocity
  • In either case, the Στ = 0

Equilibrium Equations

  • We will restrict the applications to situations in which all the forces lie in the xy plane - These are called coplanar forces since they lie in the same plane
  • There are three resulting equations

• Σ Fx = 0

• Σ Fy = 0

• Σ τ z = 0

Ch 12: Question 3

  • Can an object be in equilibrium when only one force acts upon it?

Ch 12: Question 4

  • (a) Give an example in which the net force acting on an object is zero and yet the net torque is nonzero.
  • (b) Give an example in which the net torque acting on an object is zero and yet the net force is nonzero.

Ch 12: Question 6

  • If you measure the net force and the net torque on a system to be zero, (a) could the system still be rotating with respect to you? (b) Could it be translating with respect to you?

Weighted Hand Example, cont

  • Apply the second condition for equilibrium using the joint O as the axis of rotation (Στ =0)
  • Generate the equilibrium conditions from the free-body diagram
  • Solve for the unknown forces ( F and R )

Ladder Example

  • The ladder is uniform
    • So the weight of the ladder acts through its geometric center (its center of gravity)
  • There is static friction between the ladder and the ground

Ladder Example, 2

  • Draw a free-body diagram for the ladder
  • The frictional force is ƒ = μn
  • Let O be the axis of rotation
  • Apply the equations for the two conditions of equilibrium
  • Solve the equations
    • Add a person of mass M at a distance d from the base of the ladder
    • The higher the person climbs, the larger the angle at the base needs to be
    • Eventually, the ladder may slip

Ladder Example, Extended

Ch 12: Question 12

  • A ladder stands on the ground, leaning against a wall. Would you feel safer climbing up the ladder if you were told that the ground is frictionless but the wall is rough, or that the wall is frictionless but the ground is rough? Justify your answer.

Ch 12: Problem 3

  • For what value of x will the beam be balanced at P such that the normal force at O is zero?