Unit 1 (Part A)

Part A: Mechanics of Materials

1) Centre of mass (CM) vs Centre of gravity (CG)

CM: Mass-balance point; depends only on mass distribution. Independent of gravity.
CG: Point where resultant weight acts.
Relation: In a uniform gravitational field, CM ≡ CG; in a non-uniform field, they differ.

2) Stress–strain diagram (brittle materials)

Nearly linear elastic region up to fracture; negligible plasticity. Ultimate strength ≈ fracture strength; sudden break after elastic limit. Examples: glass, ceramics, high-carbon steel (quenched).

3) Poisson’s ratio (ν)

Ratio of lateral strain to longitudinal strain within elastic limit.

ν = − (ε_lat / ε_long)

Minus sign: tension ⇒ lateral contraction. Dimensionless.

4) I-shaped girder (I-beam)

Two wide flanges joined by a thin web. Most bending stress occurs at outer fibers → flanges carry bending; web carries shear → high strength-to-weight for beams.

5) Hooke’s law

Within elastic limit, stress ∝ strain.

σ = E ε

Here E = Young’s modulus. Linear region on stress–strain plot.

7) Moment of inertia (MOI) of a rigid body

Rotational resistance about an axis; depends on mass and its distribution.

I = Σ m_i r_i² or I = ∫ r² dm

Parallel-axis: I = I_cm + M d².

8) Non-uniform bending

Bending due to localized loads (e.g., point load on simply supported beam or cantilever tip load) where bending moment and shear vary along length → curvature not constant; strain energy distribution non-uniform.

10) Couple

Two equal, opposite, parallel forces with different lines of action → zero net force but finite moment (pure rotation).

M = F × d_⊥

11) Torque (moment of a force)

Tendency of a force to rotate a body about an axis.

⃗τ = ⃗r × ⃗F , |τ| = r F sinθ

12) Elasticity

Property by which a material recovers its original shape after removal of load (within elastic limit). Beyond limit ⇒ plastic (permanent) deformation.

13) Stress (definition & unit)

Internal restoring force per unit area.

σ = F / A

SI unit: Pascal (Pa = N m⁻²). Types: tensile, compressive, shear.

14) Relation between torque and angular momentum

Rotational analogue of Newton’s 2nd law.

⃗τ = d⃗L/dt

If external τ = 0 ⇒ ⃗L is conserved.

15) Why does a wire heat on repeated bending?

Repeated bending causes internal friction and dislocation motion (plastic work). Mechanical energy dissipates as heat (elastic hysteresis) → wire becomes warm.