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Effect of Friction in Ramps

by Ron Kurtus (30 June 2016)

Resistance from friction between an object and the surface of a ramp or inclined plane reduces the ideal force mechanical advantage to a mechanical advantage including friction.

The ideal force mechanical advantage that does not consider friction is only a function of the angle of the incline. The friction between the surfaces is a function of the weight and the angle of the incline.

The force mechanical advantage including friction is found by adding the friction factor to the ideal force mechanical advantage euqation.

Questions you may have include:

This lesson will answer those questions.

Ideal force mechanical advantage

When you push an object of weight w up a ramp or inclined plane, the input force or effort required is only a function of the incline angle. The ideal force mechanical advantage of the ramp is:


MAF = 1/sin(α)


Note: α > 0 (greater than zero). If α = 0, it is no longer a ramp, and MAF is meaningless.

Pushing a load up a ramp

Pushing a load up a ramp

Friction equation

When an object is on a ramp, its weight can be broken into components, parallel and perpendicular to the surface of the inclined plane. The resulting friction equation is:

fr = μw*cos(α)


Note: The coefficient of friction, μ, is a number that is usually less than 1.0.

Relationships of ramp weight components

Relationships of ramp weight components

Mechanical advantage with friction

Friction between the ramp and the object will require a greater input or effort force to overcome the resistance. In order to obtain the desired output force for a machine with friction, the ideal input force or effort must be increased by the amount of friction. This will reduce the force mechanical advantage.

MA'F = FO/(FI + fr)


Simplify the equation by inserting values into MA'F:

FO = w

FI = w*sin(α)

fr = μw*cos(α)

MA'F = w/[w*sin(α) + μw*cos(α)]

Factor out w:

MA'F = w/w[sin(α) + μcos(α)]

Thus, for α > 0:

MA'F = 1/[sin(α) + μcos(α)]


Friction changes the force mechanical advantage when sliding an object up a ramp. To find the new mechanical advantage, you consider the ideal case with no friction, as well as the equation for friction.

The mechanical advantage of a ramp with friction is only a function of the angle of inclination of the ramp—provided the angle is greater than zero—and the coefficient of friction between the surfaces:

MA'F = 1/[sin(α) + μcos(α)]

Break down concepts to their elements

Resources and references

Ron Kurtus' Credentials


Inclined Plane - Wikipedia

Machines Resources


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Top-rated books on Simple Machines

Top-rated books on Machines

Questions and comments

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