Gravitation

8.1 Introduction

This section introduces the mechanical properties of solids by revisiting the concept of rigid body motion studied in earlier chapters, emphasizing that the motion depends on the mass distribution within the body. While rigid bodies are idealized as perfectly hard solids with fixed shape and size, real solids can deform under applied forces. For example, a steel bar can stretch or bend if a sufficiently large external force is applied, indicating solids are not perfectly rigid. The property by which a solid tends to regain its original shape and size after removal of the deforming force is called elasticity, and the deformation occurring in this process is elastic deformation. Conversely, materials like putty or mud do not regain their original shape after deformation; they undergo plastic deformation, and such materials are called plastic. The study of elasticity is crucial in engineering design, such as in buildings, bridges, automobiles, and artificial limbs, where materials must be strong yet sometimes light. The section sets the stage for understanding how forces cause deformation and how materials respond elastically or plastically.

• Rigid body motion depends on mass distribution within the body.
• Real solids can deform under applied forces; they are not perfectly rigid.
• Elasticity is the property of a material to regain original shape after deformation.
• Elastic deformation is reversible; plastic deformation is permanent.
• Plastic materials like putty do not regain original shape after deformation.
• Elastic behavior is fundamental in engineering design of structures and machines.
• 📌 Rigid body: A solid object with definite shape and size, idealized as undeformable.
• 📌 Elasticity: Property of a body to regain original shape and size after removal of deforming force.
• 📌 Elastic deformation: Reversible deformation where the body returns to original dimensions.

8.2 Stress and strain

This section defines stress and strain, fundamental quantities to describe deformation in solids under applied forces. When a body is subjected to forces but remains in static equilibrium, it may deform slightly or largely depending on the material and force magnitude. The deformation produces internal restoring forces equal and opposite to the applied forces. Stress is defined as the restoring force per unit area acting inside the body. If a force F acts normal to a cross-sectional area A, the magnitude of stress is F/A. The SI unit of stress is Newton per square meter (N/m²) or Pascal (Pa). Stress can be tensile (stretching), compressive (squeezing), or shearing (forces applied tangentially). Corresponding to these stresses, strains are defined as fractional changes in length or shape. Longitudinal strain is the fractional change in length ΔL/L under tensile or compressive stress. Shearing strain is the ratio of relative displacement Δx between opposite faces to the length L, approximately equal to the small angular displacement θ in radians. Volume strain occurs under hydraulic stress (pressure applied uniformly), defined as fractional change in volume ΔV/V. Strain is dimensionless as it is a ratio of lengths or volumes. The section also describes different types of stresses with diagrams: tensile/compressive stress elongates or compresses a cylinder; shearing stress causes angular deformation; hydraulic stress compresses a body uniformly without changing shape.

• Stress = restoring force per unit area; SI unit is Pascal (Pa).
• Tensile stress elongates a body; compressive stress shortens it.
• Longitudinal strain = ΔL/L, fractional change in length.
• Shearing stress acts tangentially, causing angular deformation.
• Shearing strain = Δx/L ≈ tan θ ≈ θ for small angles.
• Hydraulic stress (pressure) causes volume strain = ΔV/V without shape change.
• 📌 Stress: Internal restoring force per unit area within a deformed body.
• 📌 Strain: Fractional change in dimension (length, angle, or volume) due to stress.
• 📌 Tensile stress: Stress that tends to elongate the body.