Calctool's mechanical advantage calculator is a collection of straightforward and handy tools that allow you to quickly find the mechanical advantage of many simple machines. No math required!

Keep reading our article to learn:

  • What is the mechanical advantage;
  • How to calculate the mechanical advantage in various machines: the formulas for the mechanical advantage of levers, pulleys, screws, and much more;
  • Examples of the equation for the mechanical advantage in real-life situations.

Simple machines and mechanical advantage

The mechanical advantage has a simple definition: the ratio between the output force and the input force in a system. You can see that the mechanical advantage measure the force amplification of a machine.

Such simple definition, sadly, has no general equation, since the calculations for the mechanical advantage depend on the type of machine we are studying.

In the next section, we will analyze various simple machines, and we will learn what is the formula for the mechanical advantage of levers, pulleys, screws, and so on.

Here is the complete list of simple machines:

  • Lever;
  • Pulley;
  • Screw;
  • Wedge;
  • Ramp; and
  • Wheel and axle.

How to find the mechanical advantage in a lever

Levers are the simplest simple machines. Made of a fulcrum and a plank, we can identify a restricted set of quantities:

  • The effort arm lel_{\text{e}}, the extremity of the plank where we apply the input force FeF_{\text{e}}.
  • The load arm lll_{\text{l}}, the side that opposes resistance equal to the force FlF_{\text{l}}.

The fulcrum separates the load arm and the effort arm. We won't dwell on the types of levers: the ideal mechanical advantage formula doesn't change.

The equation for the force multiplication in a lever is:

MA=lellMA = \frac{l_{\text{e}}}{l_{\text{l}}}

How to calculate the mechanical advantage of a pulley

In a rope-and-pulley system, we obtain a force multiplication by changing the direction of the tension in the rope many times and adding pulleys to do so.

There aren't many things as easy as learning how to find the mechanical advantage of a lever: count the number of pulleys in your system, then apply the formula:

MA=2NpMA=2\cdot N_{\text{p}}

Where NpN_{\text{p}} is the number of pulleys. Find more about pulleys with our pulley calculator!

The mechanical advantage of a screw

Did you know that screws are simple machines too? To calculate its mechanical advantage, you need to know two parameters of the screw's thread:

  • The diameter of the screw's shaft dd; and
  • The distance traveled by the screw's shaft for each rotation (lead of the thread), ll.

The mechanical advantage is, then:

MA=πdlMA = \frac{\pi\cdot d}{l}

Mechanical advantage of a wedge

Wedges are sharp triangular objects that make splitting objects a way easier task. You can also use them to lift heavy things with less effort than using your bare hand. Since we are talking of a simple geometric shape, the formula for the ideal mechanical advantage of a wedge requires you to know only its dimensions:

  • ww — The width; and
  • ll — The length.

The mechanical advantage is nothing but:

MA=wlMA = \frac{w}{l}

Ramps: the mechanical advantage known in Ancient Egypt

The equation for the mechanical advantage of an inclined plane is similar to the one we used for the wedge.

Defining the length of the ramp (the oblique one) LL and the rise gained by the ramp VV, we can calculate the mechanical advantage with the formula:

MA=LVMA = \frac{L}{V}

Using the trigonometric functions we can also find an alternative equation:

MA=1sin(θ)MA = \frac{1}{\sin{(\theta)}}

Where θ\theta is the inclination of the ramp with respect to the horizontal.

A less known simple machine: the mechanical advantage of a wheel and axle system

A wheel and axle system allows you to multiply or divide the rotational speed (hence the torque) applied to a cylindrical object while transferring it to a smaller (or bigger) axle.

The mechanical advantage of such a system is nothing but the ratio of the radii of the components:

MA=rwraMA = \frac{r_{\text{w}}}{r_{\text{a}}}

Where rwr_{\text{w}} is the radius of the wheel, and rar_{\text{a}} is the radius of the axle. This concept is similar to the one we explored in our gear ratio calculator.

Davide Borchia
Lever
Effort arm
ft
Load arm
ft
Mechanical advantage
Pulley
Number of pulleys
Mechanical advantage
Screw
Diameter
in
Lead
in
Mechanical advantage
Wedge
Wedge length
in
Wedge thickness
in
Mechanical advantage
Ramp (inclined plane)
Ramp height
ft
Ramp length
ft
Ramp inclination
deg
Mechanical advantage
Wheel and axle
Wheel radius
in
Axle radius
in
Mechanical advantage
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