Equilibrium and elasticity formulas

Torque one way = torque other way. Pivot at hinge or unknown force. F₁ turns it one way; F₂, F₃ the other. Weight: m*9.8 at beam middle. Unused: F₃ = r₃ = 0. More: add F×r in one box (30*1.5+20*2), r = 1/1. Slant: r = dist×sin(force-to-beam angle). Ladder 53° to floor: 2.5sin(37). − answer: force points the other way. About to tip: other support = 0. Balance point: F₁ = total mass, r₁ empty, F₂ = m₁x₁+m₂x₂, r₂ = 1/1.

Up forces = down forces. Sideways: right = left (right in F₁, F₂). 2+ forces on a side: add them in one box (200+350). Unused box: 0. Weight: m*9.8. Hinge (cable T at θ), leave F₁ empty: Up: F₂ = Tsin(θ), F₃, F₄ = weights. Sideways: F₃ = Tcos(θ), F₂ = F₄ = 0. − answer: it points the other way. Hinge size and angle: vectors topic. Ladder: friction = wall force; then μ from f = μN (Forces topic).

Stretch or squeeze along the length (wire, rod, bone, column). Round: A = π(d/2)², e.g. π(0.004/2)². Hanging mass: F = m*9.8. mm: ×0.001. cm²: ×1E−4. GPa: ×1E9. Steel Y = 2.0E11, aluminum 7.0E10, copper 1.1E11, bone 1.6E10 (squeezed 9E9). Squeezed: ΔL = how much shorter (no minus). Shear (top slides sideways): S in Y, slide Δx in ΔL, height in L₀, A = face pushed. Steel S = 7.5E10, aluminum 2.5E10.

Stress = force per area (Pa = N/m²). Round wire, rod or bone: A = π(d/2)². Hanging mass: F = m*9.8. Breaking: P = breaking stress (ultimate strength). Leave F empty: biggest force. Leave A empty: smallest area. Bulk stress is just the pressure.

Strain = fraction it stretched (no unit). ×100 for %. ΔL = how much longer or shorter. Same units for ΔL and L₀. Shear strain: put Δx in ΔL (height in L₀). Know stress and modulus instead: use stress = modulus × strain.

Works for all three: Y (stretch/squeeze), S (shear), B (squeezed on all sides). Know 2 of stress, modulus, strain: get the 3rd. Type stress as F/A and strain as ΔL/L₀ if you like, e.g. 500/(π(0.002)²). Biggest strain before breaking: P = breaking stress.

Squeezed on all sides (pressure, deep water). Δp = pressure increase. ΔV comes out − (it shrinks); type ΔV with − if given. Depth h: Δp = 1000*9.8*h (sea water 1025). 1 atm = 1.013E5 Pa. Fraction ΔV/V₀: type V₀ = 1/1 (×100 for %). Water B = 2.2E9, steel 1.6E11. Compressibility = 1/B.