Simple Machines – Comprehensive Study Notes

Page 1 — Objectives

• After completing the chapter, students will be able to:

  • Describe the different types of machines (simple & compound).

  • Explain and use the terms Mechanical Advantage (MA), Velocity Ratio (VR), efficiency (η) and other related quantities.

  • Solve numerical problems based on MA, VR and efficiency.

  • Describe the concept of moment (torque) and its effects.

  • Vr, MA and effiency

Page 2 — Visual Overview of Simple Machines

• Historical banner: “40 Years of Academic Excellence (1904-2004)” – GEMS Cambridge International School.

• Infographic shows the six classical simple machines:

  1. Lever – labelled with fulcrum, load & effort.

  2. Inclined Plane – shows force applied along slope to raise load.

  3. Pulley – wheel with groove; rope lifts load.

  4. Wheel & Axle – large wheel paired with smaller axle to multiply force or speed.

  5. Wedge – double-inclined plane used for splitting/cutting.

  6. Screw – inclined plane wrapped helically around a shaft.

• Purpose of graphic: immediate recognition of each simple machine and its force-load relationship.

Page 3 — Introduction to Machines

• Definition: A machine is a device that makes human work easier, faster and more convenient.

• Four principal ways a machine helps:

  1. Changing direction of applied effort (e.g., flagpole pulley).

  2. Multiplying effort (magnifying force) so a small effort lifts a heavy load (e.g., jack screw).

  3. Increasing speed (sacrificing force for velocity, e.g., bicycle gear).

  4. Improving safety (isolating user from danger, providing stable posture, etc.).

Page 4 — Key Terminology (Part 1)

• Effort (E): Force applied to the machine in order to overcome load.

• Load (L): Resistive force lifted or overcome by the machine.

• Fulcrum (F): Fixed point about which a lever or rotating part pivots.

• Effort Arm ($ED$): Perpendicular distance from fulcrum to the point of effort application.

Page 5 — Key Terminology (Part 2)

• Load Arm ($LD$): Perpendicular distance from fulcrum to the point where the load acts.

• Input Work (Win): Work done on the machine.
  $W</em>{in}=E \times ED$

• Output Work (Wout): Work done by the machine.
  $W</em>{out}=L \times LD$

Page 6 — Principle of a Perfect Machine + Mechanical Advantage

• Principle of Machine: In a perfect (friction-less) machine,
  $W<em>{in}=W</em>{out}\;\Longrightarrow\; E \times ED = L \times LD$

• Mechanical Advantage (MA): Ratio of load overcome to effort applied.
  $MA = \frac{L}{E}$
  • Unit-less (ratio of forces).
  • Affected by friction: High friction ↓ MA.

Page 7 — Velocity Ratio & Efficiency

• Velocity Ratio (VR): Ratio of velocity/distance moved by effort to that moved by load.
  $VR = \frac{\text{Effort distance}}{\text{Load distance}}$
  • Independent of friction (purely geometrical).

• Efficiency (η): Fraction of input work converted to useful output work.
  $\eta = \frac{W<em>{out}}{W</em>{in}} \times 100\%$
  $\eta = \frac{MA}{VR} \times 100\%$
  • Practical machines have \eta < 100\% due to friction & self-weight.

Page 8 — Six Classical Simple Machines

1. Lever

2. Pulley

3. Wheel & Axle

4. Inclined Plane

5. Wedge

6. Screw

Page 9 — Pulley: Definition

• Pulley: Circular metallic/wooden disc with a grooved rim that rotates about a central axle.

• Function: Changes direction of force and/or multiplies effort depending on configuration.

Page 10 — Pulley Types Overview

1. Fixed Pulley

   • MA ≈ 1 (ideal).

   • Changes direction only.

2. Movable Pulley

   • MA ≈ 2 (ideal).

   • Magnifies effort; does not change direction.

3. Pulley System (Block & Tackle)

   • MA equals the number of rope segments supporting load (ideal).

Illustrations show ceiling-mounted fixed pulley, hand supplying effort, etc.

Page 11 — Single Fixed Pulley (Details)

• Rope passes over fixed pulley; load on one end, effort on the other.

• Effort distance = load distance ⇒ $VR=1$.

• Purpose: Only changes direction so we can pull downward to lift load upward.

Formulas for ideal fixed pulley:
$MA=1,\; VR=1,\; \eta=\frac{MA}{VR}=100\%\;\text{(theoretical)}$

Page 12 — Single Movable Pulley

• Pulley attached to load; rope anchored at one end, effort applied on free end.

• Two rope segments share the load.
  ⇒ $VR = 2$ (Effort travels twice load distance).

• Advantages: Effort is halved (ideal). Direction not changed.

Page 13 — Combined Pulley (Block & Tackle)

• Two blocks: upper (fixed) & lower (movable) each containing multiple pulleys.

• Rope woven alternately between blocks.

• Provides both direction change and effort magnification.

• Ideal rule:
  $VR = \text{Number of pulleys} = \text{Number of supporting rope segments}$

Page 14 — General VR Expression for Pulleys

$VR = \frac{\text{Effort distance}}{\text{Load distance}} = \text{No. of pulleys (or rope segments supporting load)}$

Page 15 — Numerical 1 (Worked Example)

Problem:
• A single movable pulley has efficiency $\eta = 75\%$.
• Load $L=500\,\text{N}$.
Find the required effort $E$.

Solution Outline:

1. Ideal MA for one movable pulley: $MA_{ideal}=2$.

2. Actual MA: $MA_{actual}= \eta \times \frac{VR}{100} = 0.75 \times 2 = 1.5$.

3. Compute effort:
   $E=\frac{L}{MA_{actual}}=\frac{500}{1.5}=333.3\,\text{N}$

Page 16 — Numerical 2 (Four-Pulley System)

Given:
• Four-pulley block & tackle, $\eta=80\%$, $L=1800\,\text{N}$, load raised $6\,\text{m}$.

(a) $MA<em>{ideal}=VR=4$ (four supporting segments). $MA</em>{actual}=\eta \times \frac{VR}{100}=0.80\times4=3.2$

(b) Effort:
$E=\frac{L}{MA_{actual}}=\frac{1800}{3.2}=562.5\,\text{N}$

(c) Output work:
$W_{out}=L \times h = 1800 \times 6 = 10800\,\text{J}$

(d) Input work:
$W<em>{in}=\frac{W</em>{out}}{\eta}=\frac{10800}{0.80}=13500\,\text{J}$

Page 17 — Numerical 3 (Unknown Pulleys)

Given: $\eta=80\%$, $MA_{actual}=4$, $L=1000\,\text{N}$.

1. Determine ideal VR:
   $\eta=\frac{MA<em>{actual}}{VR} \times 100\% \;\Longrightarrow\; VR=\frac{MA</em>{actual}}{\eta/100}=\frac{4}{0.8}=5$
   ⇒ Five supporting rope segments ⇒ 5 pulleys (assuming standard arrangement).

2. Effort:
   $E=\frac{L}{MA_{actual}}=\frac{1000}{4}=250\,\text{N}$

Page 18 — Wheel & Axle: Description & Theory

• Consists of two coaxial cylinders: large wheel radius $R$, small axle radius $r$ (R > r).

• Effort applied at wheel rim, load lifted at axle via rope.

• Acts as a continuous first-class lever (fulcrum at axle bearing).

• Ideal relations:
  $VR = \frac{\text{Effort distance per turn}}{\text{Load distance per turn}} = \frac{2\pi R}{2\pi r} = \frac{R}{r}$
  $MA_{ideal}=VR$

• Examples: Screwdriver, doorknob, steering wheel, sewing-machine balance wheel.

Page 20 — Wheel & Axle Numerical

Data: $R = 40\,\text{cm}=0.40\,\text{m},\; r=0.1\,\text{m},\; L=1200\,\text{N},\; E=400\,\text{N},\; \text{load rise}=4\,\text{m}$

(a) $MA = \frac{L}{E} = \frac{1200}{400}=3$
(b) $VR = \frac{R}{r} = \frac{0.40}{0.10}=4$
(c) $\eta = \frac{MA}{VR}\times100\% = \frac{3}{4}\times100\% = 75\%$
(d) Output work: $W<em>{out}=L\times h = 1200 \times 4 = 4800\,\text{J}$ (e) Input work: $W</em>{in}=\frac{W_{out}}{\eta}=\frac{4800}{0.75}=6400\,\text{J}$

Page 21 — Inclined Plane: Fundamentals

• Rigid flat surface set at angle $\theta$ to horizontal.

• Enables heavy loads to be raised with smaller effort along slope.

• Everyday examples: mountain roads, staircases, planks into trucks.

Page 22 — Inclined Plane Formulae

$VR = \frac{\text{Effort distance}}{\text{Load distance}} = \frac{\text{Length of plane }(l)}{\text{Height }(h)} = \frac{l}{h}$

If neglecting friction: $MA_{ideal}=VR$.

Page 23 — Inclined Plane Numerical Example

Given: $l=16\,\text{m},\; h=5\,\text{m},\; L=850\,\text{N},\; E=340\,\text{N}$

• $MA=\frac{L}{E}=\frac{850}{340}=2.5$

• $VR=\frac{l}{h}=\frac{16}{5}=3.2$

• $\eta=\frac{MA}{VR}\times100\%=\frac{2.5}{3.2}\times100\%=78.12\%$

• Output work (load raised vertical distance $h$):
  $W_{out}=L\times h = 850\times5=4250\,\text{J}$

Page 24 — Screw: Structure & Uses

• Screw = cylindrical shaft with helical inclined plane (thread).

• Converts rotational motion → linear motion.

• Common applications: fasteners (furniture, electronics), screw jacks (lifting cars), micrometers (precise adjustment), orthopedic bone fixation.

Page 25 — Screw Velocity Ratio

For one full turn of handle:
$VR = \frac{2\pi r}{p}$
Where:
• $r$ = radius of lever/handle.
• $p$ = pitch (axial distance between adjacent threads).

Higher VR ⇒ greater force multiplication (but greater turns for same lift).

Page 26 — Wedge

• Formed by two joined inclined planes meeting at sharp edge.

• Converts effort along length into forces normal to surfaces – splits or lifts material.

• Uses: log-splitters, axe blades, chisels, knives, doorstops.

• Velocity Ratio:
  $VR = \frac{\text{Length of wedge }(l)}{\text{Thickness / width }(w)}$

Page 27 — Compound Machines

• Compound Machine: Device combining two or more simple machines for greater overall MA.

• Each component contributes to total performance.

• Examples:
  • Bicycle – levers (brakes), wheel & axle (wheels), pulleys (chain & gears).
  • Scissors – double lever + wedge blades.
  • Wheelbarrow – lever handles + wheel & axle wheel.
  • Can Opener – levers, wedges, wheel & axle.

Ethical/Practical Insight: Compound machines reflect engineering optimisation—balancing force, speed, cost & safety to meet human needs effectively.

Cross-connections & Broader Context

• All simple machines obey energy conservation: regardless of MA, the product (force × distance) remains (ideally) constant—larger force over shorter distance or vice versa.

• Friction and material strength are the real-world constraints lowering efficiency.

• Moments (torque) underpin lever, wheel & axle, and pulley operation: $\tau = F \times d$.

• Understanding simple machines lays groundwork for analysing complex mechanical systems, robotics, biomechanics and everyday tools.