Compound Machines

Mechanical advantage

Mechanical advantage measures how much a machine or tool multiplies force, allowing you to move a load with less input force.

Mechanical advantage = output force $(F_O)$ $÷$ input force $(F_I)$

$MA=\frac{F_O}{F_I}$

Mechanical advantage does not have a unit. It is a ratio of the forces.

An MA greater than $1$ means the machine multiplies force. An MA of $1$ changes only the direction of the force, as a single fixed pulley does. An MA less than $1$ means you put in more force than the load receives, in exchange for moving the load farther or faster, as a third-class lever such as a broom does.

Example: Mechanical advantage from forces

Find the mechanical advantage of a machine with an input force of $20$ N and an output force of $100$ N.

$MA=\frac{F_O}{F_I}=\frac{100}{20}=5$

Answer: $MA=5$

You can increase mechanical advantage by increasing the distance over which the force is applied: a machine never multiplies work, so ideally input force $\times$ input distance = output force $\times$ output distance, and an MA of $5$ means you push $5$ times farther than the load moves.

In practical terms, this is lengthening a lever arm, increasing the number of pulleys in a system, or increasing the gear ratio. Pushing harder doesn’t raise mechanical advantage: a lever’s is set by the lengths of its arms, so more effort just produces proportionally more output force.

Ideal vs actual

Ideal mechanical advantage (IMA) is the theoretical mechanical advantage of a machine, assuming 100% efficiency and no energy lost to friction. IMA can be found directly from the input and output distances:

$IMA=\frac{\text{input (effort) distance}}{\text{output (load) distance}}$

Actual mechanical advantage (AMA) is the real-world mechanical advantage of a machine that accounts for friction, using the same force ratio shown earlier ($AMA=\frac{F_O}{F_I}$) with measured forces.

Efficiency: Because of friction, the work done by a machine is always less than the work done on it; therefore, machines are always less than 100% efficient. Efficiency compares the actual mechanical advantage to the ideal mechanical advantage:

$\text{Efficiency}=\frac{AMA}{IMA}\times 100\%$

Decreasing friction will increase the actual mechanical advantage of a machine.

Example: IMA and efficiency

You push a lever $10\text{ m}$ to lift a load $2\text{ m}$. Find the IMA, then find the efficiency if the measured AMA is $4$.

$IMA=\frac{\text{input distance}}{\text{output distance}}=\frac{10}{2}=5$

$\text{Efficiency}=\frac{AMA}{IMA}\times 100\%=\frac{4}{5}\times 100\%=80\%$

Answer: $IMA=5$; efficiency $=80\%$

Compound (complex) machines

A compound (complex) machine is a combination of two or more simple machines working together.

Identifying simple machines in compound machines

Bicycle: wheels (wheel and axle), handlebars (class 1 lever), gears and chain (gear/pulley system; each sprocket on its axle also acts as a wheel and axle).

Scissors: handles and blades (class 1 lever), blades (wedge)

Wheelbarrow: handles (class 2 lever), wheel (wheel and axle).

Calculating combined mechanical advantage

Combining machines multiplies force, but as before, you trade that force for distance (and speed).

To calculate combined mechanical advantage, we simply multiply the mechanical advantage of each machine together.

Total MA = $M{A}_1\times M{A}_2\times \cdots$

Example: Combined mechanical advantage

A machine has two parts. The first part has a mechanical advantage of $3$. The second part has a mechanical advantage of $4$. Find the total mechanical advantage.

Total MA = $M{A}_1\times M{A}_2$

Total MA = $3\times 4$

Answer: Total MA $=12$