Engineering Science: Simple Machines Part 1

Simple Machines Overview

A simple machine is a basic mechanical device that assists with a task by changing the magnitude (amount) or direction of a force. There are six fundamental types of simple machines:

Lever
Wheel & Axle
Pulley
Inclined Plane
Wedge
Screw

Mechanical Advantage (MA)

Mechanical Advantage is a measure of the performance of a machine, defined in two ways:

The ratio of output force to input force in a system.
The ratio of distance traveled by the input force to the distance traveled by the output force.

A mechanical advantage allows for lifting very heavy objects with a small input force. For values greater than $1$ :

A smaller effort force can move a larger resistance.
The effort force must move a greater distance than the resistance.

For values less than $1$ :

A larger effort force is needed to move a smaller resistance.
The effort force can move a shorter distance than the resistance.

Mechanical advantage can never be less than or equal to $0$ .

Distance vs. Force Trade-Off

A simple machine cannot reduce both the force required and the distance traveled simultaneously. If one value increases, the other must decrease. This maintains the product of distance and force ( $d \times F$ ) as a constant value.

Using an inclined plane with a mechanical advantage of $5:1$ as an example:

Distance: The distance traveled by the input force is $5$ times greater than the resistance. If the resistance is gravity, the length of the ramp is $5$ times greater than the vertical height would be if the object were lifted straight up.
Force: The input force (effort) required is $5$ times less than the output force (resistance). For instance, a $5\,\text{lb}$ force could push a $25\,\text{lb}$ box up the ramp.

Work

Work is the transfer of energy to an object due to an external force that moves the object from one position to another. Simple machines do not change the total amount of work required to move an object; they simply trade force for distance.

Work Formula: $W = F \cdot d_{||}$

In this formula, $F$ is force and $d_{||}$ represents the distance traveled parallel to the force that caused it.

Example Comparison: Moving a $20\,\text{lb}$ box to a height of $1\,\text{ft}$ using a ramp with an MA of $5:1$ (ramp length of $5\,\text{ft}$ , effort force of $4\,\text{lb}$ ):

Work to push up the ramp: $W = 4\,\text{lb} \times 5\,\text{ft} = 20\,\text{ft-lb}$
Work to lift straight up: $W = 20\,\text{lb} \times 1\,\text{ft} = 20\,\text{ft-lb}$

Ideal vs. Actual Mechanical Advantage

Ideal Mechanical Advantage (IMA)

IMA is the highest possible achievable mechanical advantage based on the dimensions of the machine and theoretical calculations. It does not account for friction.

IMA Formula: $IMA = \frac{D_E}{D_R}$

$D_E$ : Distance traveled by effort force (Effort Distance)
$D_R$ : Distance traveled by resistance force (Resistance Distance)

Actual Mechanical Advantage (AMA)

AMA is the mechanical advantage actually achieved in real-world application, determined through testing. It includes losses due to friction and other factors.

AMA Formula: $AMA = \frac{F_R}{F_E}$

$F_R$ : Magnitude of the resistance force (output force)
$F_E$ : Magnitude of the effort force (input force)

Efficiency

Efficiency is the ratio of useful output energy to the total input energy supplied to a system. In the context of simple machines, it can be calculated using mechanical advantage.

Efficiency Formula: $\text{Efficiency} \, (\eta) = \left( \frac{AMA}{IMA} \right) \times 100\%$

No machine is $100\%$ efficient because AMA is always less than IMA due to friction.

Levers

A lever is a rigid bar used to exert force at one point along its length by applying a force at a second point while turning about a third point on a fulcrum. A fulcrum is the fixed pivot point or support around which a lever turns.

There are three classes of levers based on the arrangement of the effort force ( $E$ ), the resistance force ( $R$ /load), and the fulcrum ( $F$ ):

First Class Lever: The fulcrum is located between the effort and the resistance. The MA can be $1$ , $< 1$ , or $> 1$ depending on the arrangement.
Second Class Lever: The resistance is located between the fulcrum and the effort force. In these levers, the effort distance ( $D_E$ ) is always greater than the resistance distance ( $D_R$ ), meaning the MA is always greater than $1$ .
Third Class Lever: The effort force is located between the fulcrum and the resistance. The effort is always closer to the fulcrum than the resistance is ( $D_E < D_R$ ), meaning the MA is always less than $1$ .

Moment and Rotational Equilibrium

Moment

A moment is a measure of the turning or rotational effect on a body about a specific axis or point due to a force acting perpendicular to the lever arm at a distance from the pivot point.

Moment Formula: $M = d \times F$

For example, a $20\,\text{lb}$ weight placed $4\,\text{in}$ from a pivot produces a moment of: $M = 4\,\text{in} \times 20\,\text{lb} = 80\,\text{in-lb}$

Rotational Equilibrium

Rotational equilibrium is a state where all rotational effects on an object are balanced, resulting in no rotation or rotation at a constant angular velocity. In this state, all moments add up to zero.

Equilibrium Condition: $M_{\text{counter-clockwise}} = M_{\text{clockwise}}$ $D_R \cdot F_R = D_E \cdot F_E$

Wheel and Axle

This machine consists of a wheel attached to a smaller axle so they rotate together, transferring force between them.

Wheel driving axle: The input force is on the wheel, causing the axle to turn.
Axle driving wheel: The input force is on the axle, causing the wheel to turn.

IMA of Wheel and Axle: The IMA can be calculated using the ratio of the radius, diameter ( $\varnothing$ ), or circumference of the wheel and axle.

If the wheel ( $10\,\text{in}$ diameter) drives the axle ( $2\,\text{in}$ diameter): $IMA = \frac{10\,\text{in}}{2\,\text{in}} = 5$
If the axle drives the wheel: $IMA = \frac{2\,\text{in}}{10\,\text{in}} = 0.2$

Pulleys

A pulley uses a wheel with a grooved rim and a rope/cable to change the direction of force or lift objects.

Fixed Pulley: $IMA = 1$ . It only changes the direction of the force.
Movable Pulley: $IMA = 2$ . The force directions are the same.
Block and Tackle (Combination Pulley System): A system using both fixed and movable pulleys with a single rope. The $IMA$ is equal to the number of supporting strands.

Compound Machines

A compound machine is a combination of two or more simple machines. The total IMA of a compound machine is the product of the individual IMAs of the machines it contains.

$\text{Total IMA} = IMA_1 \times IMA_2 \times \dots \times IMA_n$

If a system consists only of movable pulleys but they are not arranged as a block and tackle (i.e., not using a single continuous rope), it is treated as a compound machine where each individual movable pulley provides an IMA of $2$ .