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# The Mechanical Advantage Of Screw Threads

A screw is a mechanical system that converts rotational motion in to linear motion. In other worlds it converts torque (rotational force) in to a linear force.

Although ‘screw’ refers to many helical devices, in its simplest form it consists of helical threads around a cylindrical shaft (male/external thread) and its purpose is to fasten components together. At least one of these components to be fastened together contains an internal/female thread and this may be formed during the installation of the screw if the material is softer than that of the screw. This differs to the definition of a bolt which is a similarly threaded component used to fasten un-threaded components together with the use of a nut. In both cases, rotation of the threaded component, through the application of torque, results in relative movement between it and the female thread and the screw/bolt moves along its axis.

### Examples of Screw Mechanisms:

- Corkscrew
- Archimedes’ Screw
- Screw Tops for containers
- Screw Jack
- Vice

### Mechanical Advantage:

A very important property of a screw thread is that it can be used to amplify force: A small torque applied to a screw can exert a large axial force on a mass. Therefore, a threaded component is said to produce a Mechanical Advantage.

Here the resultant axial force due to an input torque is calculated from first principles.

**Screw Thread Terms:**

Pitch – The axial distance between the screw threads

Lead – The axial distance travelled by the thread during 360° revolution of the screw or nut. The smaller the lead, the higher the mechanical advantage. (Lead = Pitch X No. of Starts)

Mechanical Advantage – The ratio of axial output to rotational input force.

The ideal mechanical advantage can easily be calculated as follows:

From the conservation of energy, the work done on the screw by the rotational input force equals the work done on the mass by the resulting axial force: $$W_{in}=W_{out}$$ Work done equals the force multiplied by the distance over which it acts so, for 1 complete revolution, the work done is given by: $$W_{in}=2\pi r F_{in}$$ And the work done on the mass by the axial force is given by: $$W_{out}=l F_{out}$$

Where l = the thread lead.

Therefore, it can be seen that the ideal mechanical advantage is given by: $$MA={F_{out}\over F_{in}}={2\pi r\over l}$$ Therefore the smaller the thread lead, l, the greater the mechanical advantage and the larger the force the screw thread can exert for a given applied rotational force.

However, we know that the applied rotational force is supplied by a torque T, where: $$T_{in}=F_{in}r$$ This assumes that the lever arm for the torque is r, the radius of the thread. Substituting this in to the equation for mechanical advantage gives the resultant axial force out of the system: $$F_{out}={T_{in}2\pi \over l}$$ So it can be seen that the resultant axial force increases as the input torque increases and the thread lead decreases.

It should be noted that calculated here is the ideal mechanical advantage and does not take into account the large frictional losses due to the large area of contact between the male and female threads.

The property of a screw in that it can provide a mechanical advantage can be applied to components that utilise screw threads. Such components include Screw Jacks (also known as Jack Screw, a Worm Screw Jack, a Machine Screw Jack or a Lead Screw Jack).