# Ramps are Simple Machines

by Ron Kurtus (30 June 2016)

A ramp is an inclined plane that allows you to move an object to a greater height with less force than is needed to simply lift the object. A ramp can also be used to ease an object from a higher to a lower height.

Although it has no moving parts, a ramp is still considered a simple machine. It is called an inclined plane since it is a flat surface at an angle. The ideal force mechanical advantage (MAF) of a ramp is the weight of the object divided by the force required to move it up the ramp.

Resistance from friction can affect the mechanical advantage of a ramp. Since the friction of sliding an object up a ramp can increase the force required, rollers or wheels are usually used to reduce the friction and increase the force mechanical advantage.

Questions you may have include:

• What are the relationships of the ramp components?
• What is the idea force mechanical advantage?
• What is the mechanical advantage including inertia and friction?

This lesson will answer those questions.

## Relationships of ramp components

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

FI = w*sin(α)

FO = w

where

• FI is the input or effort force in netwtons or pounds
• w is the weight of the object
• sin(α) is the sine of angle α (small Greek letter alpha)
• FO is the load or resulting output

Note: We use small w to denote weight, while large W is work done.

Also note: The angle α must be greater than zero (α > 0). Otherwise there is no inclination, and the mechanical advantage becomes meaningless.

Pushing a load up a ramp

The force mechanical advantage of the ramp is:

MAF = FO/FI

MAF = w/w*sin(α)

MAF = 1/sin(α), provided α > 0

For example, when the angle of the ramp is α = 45°, the mechanical advantage is MAF = 1/0.707 = 1.414.

Note that speed and distance mechanical advantages are not really relevant with ramps.

## Other factors

There can be other factors that affect the force mechanical advantage of a ramp. Friction can reduce the mechanical advantage. Rollers can be used to reduce the effect of friction.

### Effect of friction

In the real world, there is friction that resists the motion.

### Components related to friction

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

Relationships of ramp weight components

If w is the weight of the object, then

• w*sin(α) is force pulling the object down the ramp
• w*cos(α) is the normal force pushing the object against the surface

The 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
• cos(α) is the cosine of angle α

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

Insert values to MA'F

FO = w

FI = w*sin(α)

fr = μw*cos(α)

MA'F = w/[w*sin(α) + μw*cos(α)]

MA'F = w/w[sin(α) + μcos(α)]

Thus:

MA'F = 1/[sin(α) + μcos(α)]

This force mechanical advantage is independent of the weight of the object but dependent on the coefficient of friction.

### Special case

When α = 0°, sin(α) = 0 and cos(α) = 1. Thus:

MA'F = 1/μ

## Summary

A ramp is an inclined plane that allows you to move a heavy object to some height with less force than needed to lift the object. A ramp can also be used to ease an object to a lower height.

Although it has no moving parts, a ramp is still considered a simple machine. It is called an inclined plane since it is a flat surface at an angle. The force mechanical advantage (MAF) of a ramp is the weight of the object divided by the force required to move it up the ramp.

Resistance factors of inertia and friction can affect the mechanical advantage of a ramp. An accelerating object requires a force to overcome its inertia. Since the friction of sliding an object up a ramp can increase the force required, rollers or wheels are usually used to reduce the friction and increase the force mechanical advantage.

Climb the ramp to success

## Resources and references

Ron Kurtus' Credentials

### Websites

Inclined Plane - Wikipedia

Machines Resources

### Books

Do you have any questions, comments, or opinions on this subject? If so, send an email with your feedback. I will try to get back to you as soon as possible.

## Students and researchers

www.school-for-champions.com/machines/
ramps.htm

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