# 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:

• What is the ideal force mechanical advantage for a ramp?
• What is the equation for the friction?
• What is the force mechanical advantage including friction?

This lesson will answer those questions.

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 = FI/FO

MAF = 1/sin(α)

where

• MAF is the ideal force mechanical advantage
• FI is the input or effort force in newtons or pounds
• FO is the load or resulting output
• α is the ramp angle of inclination (small Greek letter alpha)
• sin(α) is the sine of the ramp angle of inclination

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

## 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)

where

• fr is the resistive force of friction
• μ (small Greek letter mu) is the coefficient of friction between the two surfaces
• w is the weight of the object
• w*cos(α) is the normal force pushing the object against the surface

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

Relationships of ramp weight components

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)

where

• MA'F is the force mechanical advantage including friction
• FO is the desired output force or load
• FI is the ideal input force or effort
• fr is the resistive force of friction on the machine

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)]

## Summary

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

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Inclined Plane - Wikipedia

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