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Key Ideas

- Mechanical Advantage
- IMA
- AMA

- Efficiency
- Equilibrium
- Moment/Torque
- Machine
- Principle Machine
- Simple Machine
- Complex/Compound Machine

- Work
- Power

Mechanical Advantage

- An expression of the ratio of force output to force input
- Ideal Mechanical Advantage
- Assumes a “perfect world”
- No friction or Thermodynamics

- Distance Travelled by Effort / Distance Travelled by Load

- Assumes a “perfect world”
- Actual Mechanical Advantage
- Considers friction and Thermodynamics
- Force applied by Load / Force applied by Effort

- Efficiency
- A measure of the useable portion of energy in a system
- AMA / IMA

Equilibrium

- Assumes a “perfect world”
- Efficiency = 1
- AMA = IMA
- DEFE = DLFL
- FE:FL = DL:DE
- Ratio of Forces is INVERSE of Ratio between Distances

Lever

- Beam (LEVER ARM) supported by pivot point (FULCRUM)
- 3 classifications
- One of two PRINCIPLE MACHINES
- Force Multiplier or Distance Multiplier
- “Give me a lever long enough and a fulcrum on which to place it, and I shall move the world.” Archimedes

Class 1 Lever

- Fulcrum between Load and Effort
- EFL

Class 2 Lever

- Load between Fulcrum and Effort
- FLE

Class 3 Lever

- Effort between Fulcrum and Load
- FEL

Wait a “moment!”

- Moment: a measure of the force inducing the tendency of an object to rotate within a system.
- measured by the application of a force some distance from the “center of rotation”

- This is virtually the same concept as “Torque”
- This is NOT the same thing as “Torsion,” the structural stress resulting from moment/torque
- Torque = Moment = F * D = τ
- (that’s a lower-case Greek letter, “tau.”)

- Measured (USCMS) in Foot-Pounds (ftlbs)

Lever Equilibrium

- D = Distance travelled by Force
- Assume rotation doesn’t stop
- D = pi*2*radius (distance from fulcrum to force)

- => dEFE = dLFL
- Distance between Effort and Fulcrum * Force of Effort
- Distance between Load and Fulcrum * Force of Load
- Compare these equations to “Moment”

- => dE:dL = hE:hL
- Height travelled = d sin ß
- ß is the same for both sides of the lever, so…
- dE sin ß = dL sin ß
- Therefore dE = dL<<implies>>hE = hL

Ideal Mechanical Advantage

- Theoretical Mechanical Advantage
- Levers can be FORCE MULTIPLERS or DISTANCE MULTIPLERS
- IMA of a Lever: dE / dL
- >1 - Force Multiplier
- =1 - neutral system
- <1 - Distance Multiplier

Wheel-and-Axle

- Behaves as Class 2 Lever
- ONLY WHEN EFFORT IS APPLIED TO WHEEL!!!!!!!!!

- Behaves as Class 3 Lever
- WHEN EFFORT IS APPLIED TO AXLE!!!!!!!!!

- Force Multiplier, distance reducer
- (steering wheel)

- Distance Multiplier, force reducer
- (automotive transmission)

Wheel & Axle

- D = Distance travelled by Force
- D = pi*2*radius (distance from CoR to force)
- D = pi*diam. = pi*2*rad. = Circum

- => dEFE = dLFL
- Distance between Effort and CoR * Force of Effort
- Distance between Load and CoR * Force of Load
- Compare these equations to “Moment”

Pulley

- Grooved wheels attached to an axle
- Grooves runs concentrically around the outer rim of the wheel
- Behave like Class 2 Levers
- Direction Changer, Force Multiplier, or Distance Multiplier
- “Open” system or “Closed” system
- DE measured by length of rope
- DL measured by lift of load

Pulley as Direction Changer

- Open pulley systems leave disconnected the ends of the rope/cable/chain/belt

IMA of Fixed Pulley: 1

Pulley as a Force Multiplier

IMA of fixed pulley: 1

IMA of moving pulley: 2

IMA = 4?!!?

AH!! 2 Pulleys!

Compound Machines

- When two or more simple machines are used in conjunction with one another
- Can be same machine (pulleys and pulleys)
- Can be different machines (lever, w/a, pulley)

- Total IMA = Product of simple IMA
- MAT = MA1 * MA2 * … * MAn

Closed Pulley Systems

- Closed pulley systems have connected the ends of the belt/cable/chain/cable
- Behave somewhat like a wheel-and-axle… just in two pieces

Follower

Load

Resistance

Output

Driver

Effort

Input

Therefore…

- SEVERAL equivalent equations!!
- New Variables!!
- d = diameter
- τ = torque
- ω = Rotational Velocity (rotations-per-minute; revolutions-per-minute; RPM)

- IMA = dout/din = ωin/ωout
- AMA = τout/τin

Inclined Plane

- Second PRINCIPLE MACHINE
- Reduces the force required to lift an object
- Ideal Mechanical Advantage: length of slope / height of slope
- NOT THE SAME AS CALCULATION OF SLOPE ANGLE
- NOT A MOVING OBJECT!

Length of Slope

Height

Wedge

- Basically two inclined planes connected
- Functions as moving IP

Length of Slope

Length of Slope

½ Face

Face

Therefore…

- EQUATION FOR Wedge EQUILIBRIUM
- 2sE = fL
- 2 * Length of Slope * Force of Effort
- Width of Wedge Face * Force of Load

- EQUATION FOR PULLEY MECHANICAL ADVANTAGE
- 2s / f
- 2 * Length of Slope / Width of Wedge Face

Screw

- Theoretical Mechanical Advantage: pi*dm / l
- pi = (appx.) 3.1415 or 22/7
- dm = average diameter of the screw
- l = “lead” of the screw
- axial advance of a helix for one complete turn on a gear
- In other words… the distance between threads

Gears

- Same basic idea as Pulleys
- Gears have teeth or spurs extending radially outward from the outer or inner edge of the wheel
- Gears do not slip, as pulleys can
- Gears ALWAYS reverse the direction of rotation between adjacent gears
- Use an “idler gear” between driver and follower to have follower turn in same direction as driver

- Force Multiplier or Speed Multiplier

Therefore…

- SEVERAL equivalent equations!!
- New Variables!!
- d = diameter
- τ = torque
- ω = Rotational Velocity (rotations-per-minute; RPM)
- n = number of teeth

- IMA = nout/nin = dout/din = τout/τin = ωin/ωout
- IMA = “GEAR RATIO”

Arbeit macht frei

- WORK = FORCE x DISTANCE
- In a way, measures the conversion of “POTENTIAL ENERGY” into “KINETIC ENERGY”
- No distance = no work.
- No force = no work.
- TORQUE = rotational work
- TORQUE = FORCE x RADIUS

She can’t do it, Captain! I need more power!

- Power = Work / Time
- Horsepower (hp) = (Force in pounds x Distance in feet) / (Time in seconds x 550)
- Yep… the number (constant) 550…
- HP was originally used by James Watt to describe the “power” equivalence of steam engines in terms we could understand
- This number was chosen… for some reason… but it’s actually twice the number that it should be… the first motor was THAT powerful…
- Electrical Power is measured in WATTS
- 1 Watt = 1 Joule / 1 Second

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