Simple Machine Notes - POE 1.1

*Some of the formulas are messed up but they are stated in other parts of the note

Six Simple Machines: Overview

• The six simple machines are: Lever, Wheel and Axle, Pulley, Inclined Plane, Wedge, and Screw. Mechanisms manipulate magnitude of force and distance to achieve work more efficiently.

• This transcript focuses on Lever, Wheel and Axle, and Pulley, with references to the other three (Inclined Plane, Wedge, Screw) as part of the six basic machines.

Mechanical Advantage (MA) and Related Concepts

MA definitions

MA is divided into 2 categories IMA and AMA. IMA assumes ideal conditions while AMA accounts for friction.

• Ideal Mechanical Advantage (IMA)

  • Theory-based calculation that neglects friction

  • IMA = ratio of distances moved by the effort and resistance: IMA = $\frac{D_{E}}{D_{R}}$

  • In terms of distances around a lever or wheel: DE = distance traveled by effort, DR = distance traveled by resistance (circular paths for levers and wheels)

• Actual Mechanical Advantage (AMA)

  • Real-world MA that accounts for friction and losses

  • AMA = ratio of magnitudes of resistance to effort: AMA = $\frac{F_{R}}{F_{E}}^{}$

  • Used in efficiency calculations

• Efficiency

  • Defined as the ratio of useful output to total input, equals the ratio of AMA to IMA: $\eta = \frac{AMA}{IMA}$

  • No machine is 100% efficient; losses due to friction, heat, etc.

• Work

  • Work is force applied on an object times the distance the object travels parallel to the force: $W = F \cdot d_{\parallel}$

  • The product of the effort force and the distance moved is the same for all configurations of a given machine if ideal (ignoring friction), but actual work losses occur due to friction.

Work, Distance, and Perpendicular Components

• When moving, the effective distance is the component parallel to the force: $W = F \cdot d_{\parallel}$

• In many diagrams, initial and final positions are used to illustrate the distance moved and the work done.

Lever: Definitions and Classifications

• A lever is a rigid bar that exerts pressure or sustains a weight by applying a force at one point, turning on a fulcrum at a second point.

Lever classes (based on fulcrum position relative to effort and resistance)

• First Class Lever

      Fulcrum is between the effort and the resistance forces.

      Both effort and resistance forces are applied to the lever arm in the same direction.

      Can have MA > 1 or MA < 1 depending on distances; MA can equal 1 in a     balanced arrangement.

      Diagram notes: MA = 1 in a balanced case; MA < 1 when load is far from fulcrum     relative to effort; MA > 1 when load is close to fulcrum compared to effort.

• Second Class Lever

  • Fulcrum located at one end; Resistance force is between the fulcrum and the effort force.

  • Resistance and effort forces act in opposite directions.

  • Always has MA > 1 (effort force is amplified).

• Third Class Lever

  • Fulcrum located at one end; Effort force is between the fulcrum and the resistance.

  • Resistance and effort forces act in opposite directions.

  • Always has MA < 1 (distance advantage is lost; greater effort is required for the same load).

• Moment (Torque)

  • The turning effect of a force about a point:

  • Definition: $M = d \times F$ (perpendicular distance times force)

  • Also called torque: a force that produces or tends to produce rotation or torsion.

• Lever moment calculations (example)

  • Given: Effort = 15 lb, distance (fulcrum to effort) = 5.5 in.

  • Moment: M_E = d \cdot F = 5.5\text{ in} \cdot 15\text{ lb} = 82.5\text{ in·lb}

  • If the resistance moment is known, rotational equilibrium occurs when the sum of all moments on the lever equals zero: clockwise moments = counterclockwise moments.

Lever: IMA, AMA, and Distances

• IMA for a lever

  • Circular motion concept: DE = distance traveled by effort = $DE = 2\pi (\text{effort arm length})$

  • DR = distance traveled by resistance = $DR = 2\pi (\text{resistance arm length})$

  • Therefore: $IMA = \frac{DE}{DR} = \frac{2\pi (\text{effort arm length})}{2\pi (\text{resistance arm length})} = \frac{\text{effort arm length}}{\text{resistance arm length}}$

• AMA for a lever

  • AMA is the ratio of load to effort: $AMA =$ $\frac{F_{R}}{F_{E}}$

  • Example values: with effort arm = 5.5 in and resistance arm = 2.25 in, if FE = 16 lb and FR = 32 lb, then

  • $AMA=\frac{F_{R}}{F_{E}}$ = 2:1

• Relation between AMA and IMA

  • In example: AMA = 2:1, IMA ≈ 2.44:1 (illustrative)

  • Why IMA can be larger than AMA: due to frictional losses and non-idealities in real systems

• Efficiency for a lever system

  • Efficiency η = $\frac{AMA}{IMA}$

  • In the example: AMA = 2:1, IMA = 2.44:1, so η ≈ 0.82 (82%) (illustrative; actual efficiency depends on friction and losses)

• Real-world lever considerations

  • Real lever systems have inefficiencies; no lever is perfectly efficient

Wheel and Axle

• Structure

  • A wheel is a lever arm fixed to a shaft (the axle); they move together as a simple machine to lift/move objects by rolling.

  • Important to identify which component is applying the effort and which is the resistance force (wheel vs. axle).

• IMA for wheel and axle

  • Distances traveled by effort and resistance are circular: DE = circumference traveled by the effort, DR = circumference traveled by the resistance.

  • Formulas:

  • $DE = 2\pi (\text{effort radius}) = \pi (\text{effort diameter})$

  • $DR = 2\pi (\text{resistance radius}) = \pi (\text{resistance diameter})$

  • Therefore: $IMA = \frac{DE}{DR} = \frac{\pi D<em>E}{\pi D</em>R} = \frac{D<em>E}{D</em>R}$

  • Example: If the wheel diameter is 20 in and the axle diameter is 6 in, then

  • IMA = $\frac{D_{E}}{D_{R}}$ = approx 3.33:1

  • If the axle drives the wheel (reversed), IMA = $\dfrac{6}{20} = 0.3:1$

• AMA for wheel and axle

  • AMA = $\frac{F<em>R}{F</em>E}$ (load to effort)

  • Example: Given a wheel-axle system with example forces, AMA ≈ 2.86:1 (from a typical illustration: 200 lb load vs 70 lb effort)

• Efficiency

  • $\eta = \frac{AMA}{IMA}$

  • Example efficiency: ≈ 85.9% for the illustrated wheel-axle system

Pulleys

• Definition

  • A pulley is a lever with a wheel that has a groove for a rope or cable to change the direction and magnitude of a force.

• IMA for pulleys

  • Fixed pulley (first class lever with IMA = 1)

  • Changes the direction of the applied force but does not multiply effort: $IMA = 1$

  • Movable pulley (second class lever with IMA = 2)

  • Provides a mechanical advantage by sharing the load; direction of forces remains the same in the rope configuration

• Pulleys in combination (block and tackle)

  • Fixed and movable pulleys combined to provide greater MA and direction change

  • IMA is equal to the number of strands opposing the load (assuming ideal rope, ignoring friction and angle effects): for a system with N strands opposing the load, $IMA = N$

  • If a single rope threads multiple times through a system, IMA increases with the number of supporting strands

• Compound pulley systems and rope configurations

  • When separate ropes or cables connect, the output of one pulley system can become the input to another, forming a compound machine

  • Example IMA problems may involve multiplying individual IMAs: $IMA<em>{total} = IMA</em>{pulley} \times IMA_{lever}$ (illustrative)

• AMA for pulleys

  • AMA $=\frac{F_{R}}{F_{E}}$

  • Example: A pulley system with load 800 lb and effort 230 lb yields $AMA = \frac{800}{230} \approx 3.48:1$

• Efficiency for pulley systems

  • $\eta = \frac{AMA}{IMA}$

  • Example efficiency: ≈ 87% for the illustrated pulley system

• Common considerations and misconceptions

  • IMA for pulleys depends on the number of strands opposing the load; it is not always the count of strands pulling in any single direction

  • Angles and geometry affect IMA; trigonometry may be required for non-ideal pulley configurations

  • A common misconception: simply counting strands or directions without considering load direction and angle can lead to incorrect IMA values

• Pulley IMA details

  • IMA = # strands opposing the load when strands are oriented to oppose the load

  • In some diagrams, the IMA can be represented as 2, 3, 4, etc., depending on the configuration

Real-World and Design Considerations

• Real-world MA questions posed in the slides

  • Can you think of a machine with MA > 1? (Yes—most levers and pulleys with advantage)

  • Can you think of a machine with MA < 1? (Yes—some levers, certain gear setups, where the user must apply more force to achieve small loads)

• Real-world brands cited as examples of MA concepts

  • CISEX, VE, CYBEX (examples used to illustrate machines with MA > 1 or < 1 in real contexts)

Formulas and Quick Reference (LaTeX)

• Work:

  • $W = F \cdot d_{\parallel}$

• Ideal Mechanical Advantage (IMA):

  • IMA = $\frac{D_{E}}{D_{R}}$

• Actual Mechanical Advantage (AMA):

  • $AMA=\frac{F_{R}}{F_{E}}$

• Efficiency:

  • $\eta = \frac{AMA}{IMA}$

• Moment (Torque):

  • $M = d \cdot F$

• Wheel and Axle IMA:

  • $IMA = \frac{D<em>E}{D</em>R}$ (or equivalently $IMA = \frac{\text{diameter of effort}}{\text{diameter of resistance}}$)

• Wheel and Axle AMA:

• Pulley IMA:

  • For ideal configurations, $IMA = \text{number of strands opposing the load}$

• Pulley AMA:

  • $AMA = \frac{F<em>R}{F</em>E}$

• Compound machines (IMAtotal):

  • $IMA<em>{total} = \prod IMA</em>i$

  • Example: $IMA<em>{total} = IMA</em>{pulley} \cdot IMA_{lever}$

• Efficiency relation for pulleys and levers:

  • $\eta = \frac{AMA}{IMA}$

Notes on Notation and Conventions

• DE and DR denote distances traveled by the effort and resistance, respectively, in IMA calculations for levers and wheels.

• Distances for circular motion use circumferences: $DE = 2\pi r<em>E = \pi D</em>E$ and $DR = 2\pi r<em>R = \pi D</em>R$.

• Distinguish between IMA (ideal, frictionless) and AMA (actual, friction-included) when evaluating system performance.

• When calculating unknowns from rotational equilibrium, ensure units are consistent (e.g., in-lb for moments).

Quick Practice Prompts (conceptual)

• If a lever has the resistance arm twice as long as the effort arm, what is the MA direction and magnitude tendency?

• In a wheel-and-axle system where the wheel diameter is 20 in and the axle diameter is 6 in, what is the IMA when the wheel is the effort and the axle is the resistance? And what is the IMA if the axle drives the wheel?

• How does adding a movable pulley in a block-and-tackle configuration affect IMA relative to a single fixed pulley?

• If AMA = 3.0 and IMA = 4.5 for a pulley system, what is the efficiency?

• For a lever in rotational equilibrium with a known effort moment, how would you determine the unknown resistance distance from the fulcrum?

Real-World Relevance and Implications

• Understanding MA, IMA, and AMA helps in design optimization: achieving required motion and force without overengineering.

• Inefficiencies due to friction must be accounted for in real designs, influencing material choice, lubrication, and mechanical tolerances.

• Ethical and practical implications: selecting appropriate mechanisms reduces user effort, increases safety, and can impact accessibility and ergonomics in tools and assistive devices.

Connections to Foundational Principles

• Energy conservation: the work input (accounting for losses) equals work output; efficiency captures energy losses.

• Newtonian mechanics: forces, moments, and torques determine motion and equilibrium of lever systems.

• Geometry and trigonometry (for angles in pulley systems): accurate IMA requires proper accounting of angle effects in non-ideal configurations.