chapter11mine-1

Equilibrium and Elasticity

Chapter 11 Overview

• Introduction to the principles of equilibrium and elasticity within physical systems.

Conditions for Equilibrium (11-1)

• Static Equilibrium: An object is either at rest or moving with constant velocity.

  • Applies to both linear and rotational motion in an inertial reference frame.

  • First Condition: The vector sum of all external forces must be zero:

    • (\Sigma F = 0)

  • Second Condition: The sum of all external torques must be zero about any point:

    • (\Sigma \tau = 0)

Detailed Conditions for Equilibrium

• Net Force Equal to Zero:

  • For the center of mass of a body to remain at rest, the net external force must equal zero: ( F = 0 )

• Net Torque Equal to Zero:

  • For a nonrotating body, the net external torque around any point must equal zero: ( \Sigma \tau = 0 )

Types of Equilibrium

• Translational Equilibrium:

  • Defined by the condition that translational acceleration of the object's center of mass is zero. Applicable in inertial reference frames.

• Rotational Equilibrium:

  • Defined by the condition that angular acceleration is zero for any axis of rotation.

Equilibrium Equations

• Applications limited to situations with forces in the x-y plane.

• Results in three scalar equations to solve equilibrium problems:

  • ( \Sigma F_x = 0 )

  • ( \Sigma F_y = 0 )

  • ( \Sigma \tau = 0 )

Center of Gravity

• Concept:

  • Represents the point where equivalent total gravitational force acts on a body.

  • Each mass element exerts a torque equal to its weight times its moment arm.

• Single Point: Treats weight as acting through the center of gravity (CG) when gravity variation is negligible.

• Equilibrium Condition: In rotational equilibrium, if supported at a point, CG is directly above or below that point.

  • Holding object at CG results in zero torque due to its weight.

Multiple Support Points

• Requirement for Equilibrium: A body must have its center of gravity within the area bounded by its supports.

  • If CG lies outside, the body is not in equilibrium.

Solving Rigid-Body Equilibrium Problems (11-3)

• Step-by-Step Process:

  1. Identify the first and second conditions for equilibrium: ( \Sigma F_x = 0 ), ( \Sigma F_y = 0 ), and ( \Sigma \tau = 0 ).

  2. Sketch the physical setup and determine the analyzed body.

  3. Select a coordinate system for linear and rotational motion.

  4. Draw a free-body diagram outlining all forces present.

  5. Choose a reference point for computing torques.

  6. Write equations reflecting equilibrium conditions; if necessary, use multiple reference points for torques.

Strain, Stress, and Elastic Moduli

• Properties of Solids:

  • Solids maintain shape but can deform.

• Stress: Force applied per unit area.

• Strain: Resultant deformation from applied stress.

  • For small stresses, the relationship between stress and strain is linear, following Hooke's law.

Elastic Modulus

• Definition: Constant of proportionality between stress and strain, characteristic of the material and deformation type.

  • Relates applied stress to the material response.

• Deformations:

  • Elastic Deformation: Reversible.

  • Plastic Deformation: Irreversible.

• Excessive stress leads to material failure.

Tensile Stress and Strain

• Tensile Stress: Measure of force to cross-sectional area; force is perpendicular to the area.

• Tensile Strain: Ratio of change in length to original length.

Young’s Modulus

• Characterizes linear elastic behavior within small tensile stress ranges; defined as stress over strain:

  • Units: N/m².

• Characteristics: High values indicate stiff materials (e.g., Steel: ( 20 \times 10^{10} ) N/m²).

• Compression: The same principles apply in reverse when forces are applied.

Bulk Stress and Strain

• As surrounding pressure changes, solid volume alters accordingly.

• Bulk Modulus: Ratio of volume stress to volume strain; negative value indicates that increased pressure decreases volume.

Shear Stress and Strain

• Shear Deformation: Changes the shape of the object due to tangential forces.

• Shear Stress: Defined as force per area.

• Shear Strain: Horizontal distance moved to height ratio (( \Delta x / h )).

• Shear Modulus: Ratio of shear stress to shear strain; units are N/m².

Elasticity and Plasticity

• Hooke’s Law: Proportionality between stress and strain is limited to certain ranges of deformation.

• Elastic Hysteresis: A phenomenon depicted in stress-strain diagrams, particularly in materials like vulcanized rubber.