
How do you drive a car up to the fifth level of a city parking garage? Chances are good that there is not enough space around the parking garage to fit a straight, quartermile long ramp! Instead, architects will design some parking garage ramps to spiral around. You get off the ramp when you get to the fifth floor. Parking ramps such as the one at right can be wound around the shape of a cylinder to conserve space in a densely populated city environment.

You have used many of these spiral ramps before, although you might not have realized it!
The threads of a screw resemble a spiraling ramp. The threads wrap around a central cylinder, just as the parking garage ramp in the illustration wraps around a central cylinder. You can make the shape of a spiral ramp yourself using a triangular piece of paper wrapped around a pen. Ramps, wedges, and screws are therefore all related simple machines: They all use the principles of an inclined plane.

The key property of a screw is its pitch p, which is the vertical distance between each successive thread. When you turn the screw around once, the screw will travel a distance p into the material. The output work done is then F_{o}×p. What is the screw’s mechanical advantage? The input force is usually applied to the screw using another device, such as a screwdriver. When your hand has turned the outside of the screwdriver handle once, it has traveled a circumference of 2πL, where L is the radius of the handle. The input work done is therefore F_{i}×(2πL). Since the input work equals the output work, the mechanical advantage of the screw can be derived:
$$\begin{array}{ccccc}{F}_{i}\times (2\pi L)={F}_{o}\times p& \Rightarrow & \frac{{\overline{)F}}_{i}\times (2\pi L)}{{\overline{)F}}_{i}\times p}=\frac{{F}_{o}\times \overline{)p}}{{F}_{i}\times \overline{)p}}& \Rightarrow & M{A}_{screw}=\frac{{F}_{o}}{{F}_{i}}=\frac{2\pi L}{p}\end{array}$$

(12.9)  $$M{A}_{screw}=\frac{2\pi L}{p}$$
 MA_{screw}  =  mechanical advantage  L  =  radius of screwdriver handle (m)  p  =  pitch of screw (m) 
 Mechanical advantage of a screw 

Francine is using a screwdriver with a handle with a diameter of 1.5 in to drive in a 1/420 screw (one that has 20 threads/in). What is the mechanical advantage of her screw and screwdriver combination?

The mechanical advantage is 94. The radius of the screwdriver handle is onehalf of its diameter or L = ½(1.5 in) = 0.75 in. The screw has 20 threads/in, so its pitch is 1/20th of an inch or p = 0.05 in. The mechanical advantage is therefore
$$M{A}_{screw}=\frac{2\pi (0.75\text{in})}{0.05\text{in}}=94$$
