Machines

Overview

Machines focuses on simple machines, mechanical advantage (MA), efficiency, and power. Master free‑body diagrams (FBDs), torque balance, multi‑stage systems, and error‑aware calculations.

Simple machines and relations

• Levers: classes I–III by fulcrum/effort/load positions. Ideal mechanical advantage (IMA) ≈ effort arm / load arm. Torque balance Στ = 0 at equilibrium.
• Pulleys: fixed vs movable; block‑and‑tackle. IMA equals the number of supporting rope segments (ideal). Watch for non‑vertical segments.
• Gears: gear ratio = teeth_driven / teeth_driver = ω_driver / ω_driven = τ_driven / τ_driver (ideal). Idlers change rotation direction only.
• Wheel and axle: IMA ≈ radius_wheel / radius_axle (ideal). Friction in bearings reduces efficiency.
• Inclined plane: IMA ≈ length / height; with friction, effort increases by μN components.
• Screws: IMA ≈ 2πr / pitch (ideal). Include thread friction for realistic torque estimates.

Efficiency and power

• Actual MA (AMA) = load / effort. Efficiency η = AMA / IMA = (W_out / W_in). Losses from friction, deformation, and misalignment.
• Power P = W / t = τω for rotation; average vs instantaneous distinctions.

Multi‑stage systems

• Cascading IMA: multiply stage IMAs (ideal), then apply stage efficiencies to estimate AMA.
• Mixed systems: translate pulley output force into lever input, or gear output torque into winch lifting, etc.; keep directions and sign conventions consistent.

Free‑body diagrams and torque

• Draw forces at correct locations; decompose into components along useful axes; choose pivot to simplify (often at unknown reaction to eliminate it).
• Common supports: pin (Rx,Ry), roller (Ry), smooth surface (normal only), rough surface (normal + friction up to μN).

Worked micro‑examples

1. Compound pulley with non‑vertical leg

• A 4‑segment support ideally gives IMA=4. If one leg is at 30° to vertical, effective support on that leg = T cos 30°. Sum vertical components to compute effort for a given load and adjust AMA vs IMA.

2. Lever with angled force

• Effort F applied at angle θ to lever arm length L_e; torque = F L_e sin θ. An oblique pull reduces effective torque; recalc required effort for load τ_load at arm L_l.

3. Gear train with belt stage

• Driver gear 12T → driven 36T (3:1) → belt to pulley with diameters 80:40 (2:1 speed‑up). Net speed ratio = 3/2, torque ratio = 2/3 (ideal). With η_gears=0.9, η_belt=0.85, overall η ≈ 0.765.

4. Incline with friction

• Load W up incline angle α with μ_k. Required force F ≈ W (sin α + μ_k cos α). IMA ideal = L/H = 1/sin α; AMA smaller due to friction.

5. Screw jack torque

• To lift load W with screw pitch p and handle radius r (ideal): input work per turn 2πr·F_in = output rise p·W → F_in ≈ (p W)/(2π r η). Include efficiency.

Measurement and uncertainty

• Propagate uncertainty for products/quotients by adding relative uncertainties; for sums/differences add absolute uncertainties. Report with appropriate sig figs.
• Calibrate spring scales and distances; note zero offsets and parallax.

Pitfalls

• Counting supporting segments incorrectly in pulley systems with redirecting anchors.
• Ignoring lever arm orientation (using L instead of L sin θ for torque).
• Mixing gear tooth counts with diameter ratios inconsistently.
• Assuming 100% efficiency when data clearly show losses; not distinguishing AMA from IMA.
• Dropping units or mixing N and kgf.

Practice prompts

• Draw FBDs and compute required effort for a mixed lever–pulley system with given μ and angles.
• Design a gear train to achieve 5:1 torque increase within size constraints; compute output speed and discuss efficiency impacts.
• Given measured input/output forces and distances, compute AMA, IMA, and η with uncertainty.

References

• SciOly Wiki – Machines: https://scioly.org/wiki/index.php/Machines