
Consider using a board and a log to lift a heavy boulder. The board and log together create a simple machine (a lever) that converts your input force into a more powerful output force capable of lifting the boulder. As you apply an input force downward on one end of the board, this lever applies a larger output force upward on the boulder at the other end. In terms of forces, the lever both magnifies the input force and changes its direction.

The mechanical advantage tells you the “magnification factor” relating the strength of the output force to the strength of the input force. Mathematically, the mechanical advantage (MA) is the output force divided by the input force. A mechanical advantage of 5 means that the output force is 5 times larger than the input force.

(12.1)  $$MA=\frac{{F}_{o}}{{F}_{i}}$$
 MA  =  mechanical advantage  F_{o}  =  output force (N)  F_{i}  =  input force (N) 
 Mechanical advantage


When MA = 1, the magnitude of the output force is equal to the input force—the machine has no mechanical advantage. A mechanical advantage greater than one (MA > 1) means the output force is larger than the input force. Mechanical advantage can also be less than one! The overall mechanical advantage of a bicycle is less than one because a bicycle trades lower output force for greater output distance.

Inventor Prince Ludwig designed a handcranked machine to lift water from a stream to the kitchen window. Igor, Ludwig’s assistant, counted 27 gears, 7 levers, 12 wheels, and 3 ramps in the machine. Igor found that the machine required 200 N of applied force to lift 100 kg of water. What is the mechanical advantage of Prince Ludwig’s machine?
Asked: 
mechanical advantage MA 
Given: 
input force F_{i} = 200 N;
mass of water lifted, m = 100 kg 
Relationships: 
F_{w} = mg,
MA = F_{o}/F_{i} 
Solution: 
The output force that is lifted is the weight of the 100 kg of water:
$${F}_{w}=mg=(100\text{kg})(9.8{\text{m/s}}^{2})=980\text{N}$$
The mechanical advantage of Prince Ludwig’s contraption is therefore
$$MA=\frac{{F}_{o}}{{F}_{i}}=\frac{980\text{N}}{200\text{N}}=4.9$$

Answer: 
The mechanical advantage is 4.9.
Note that mechanical advantage compares the input and output forces, regardless of how the machine is constructed inside! 

Jaylene has a mass of 60 kg and she has a simple lever with a mechanical advantage of 4. What is the maximum weight of an object that she can lift when she stands on one end of the lever?

The answer is 2,350 N.
The input force for the lever is her weight F_{i} = F_{w} = mg. The maximum output force is therefore
$$\begin{array}{ccc}MA=\frac{{F}_{o}}{{F}_{i}}& \Rightarrow & {F}_{o}={F}_{i}\times MA=(60\text{kg})(9.8{\text{m/s}}^{2})\times 4=\mathrm{2,350}\text{N}\end{array}$$
