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# Inclined plane

## Uses

Inclined planes are widely used in the form of loading ramps to load and unload goods on trucks, ships, and planes. Wheelchair ramps are used to allow people in wheelchairs to get over vertical obstacles without exceeding their strength. Escalators and slanted conveyor belts are also forms of inclined plane. In a funicular or cable railway a railroad car is pulled up a steep inclined plane using cables. Inclined planes also allow heavy fragile objects, including humans, to be safely lowered down a vertical distance by using the normal force of the plane to reduce the gravitational force. Aircraft evacuation slides allow people to rapidly and safely reach the ground from the height of a passenger airliner. Other inclined planes are built into permanent structures. Roads for vehicles and railroads have inclined planes in the form of gradual slopes, ramps, and causeways to allow vehicles to surmount vertical obstacles such as hills without losing traction on the road surface. Similarly, pedestrian paths and sidewalks have gentle ramps to limit their slope, to ensure that pedestrians can keep traction. Inclined planes are also used as entertainment for people to slide down in a controlled way, in playground slides, water slides, ski slopes and skateboard parks.

## Terminology

### Slope

The mechanical advantage of an inclined plane depends on its slope, its gradient or steepness. The smaller the slope, the larger the mechanical advantage, and the smaller the force needed to raise a given weight. A plane's slope s is equal to the difference in height between its two ends, or "rise", divided by its horizontal length, or "run".{{cite book | last = Handley | first = Brett | authorlink = |author2=David M. Marshall |author3=Craig Coon | title = Principles of Engineering | publisher = Cengage Learning | year = 2011 | location = | pages = 71–73 | url = https://books.google.com/books?id=3YBeXkp-AacC&pg=PT91&lpg=PT91&dq=%22inclined+plane%22+slope+angle+%22mechanical+advantage%22&source=bl&ots=fv4F2vb0uT&sig=gJBSYxtVOdAU-RovppyExeG9tTA&hl=en&sa=X&ei=3TJMUNT3KM3biwKf74HYAg&ved=0CDEQ6AEwAA#v=onepage&q=%22inclined%20plane%22%20slope%20angle%20%22mechanical%20advantage%22&f=false | doi = | id = | isbn = 978-1-4354-2836-2}} It can also be expressed by the angle the plane makes with the horizontal, θ. \theta = \tan^{-1} \bigg( \frac {\text{Rise}}{\text{Run}} \bigg) \,

## Inclined plane with friction

### Analysis

A load resting on an inclined plane, when considered as a free body has three forces acting on it:
• The applied force, Fi exerted on the load to move it, which acts parallel to the inclined plane.
• The weight of the load, Fw, which acts vertically downwards
• The force of the plane on the load. This can be resolved into two components:
• *The normal force Fn of the inclined plane on the load, supporting it. This is directed perpendicular ( normal) to the surface.
• *The frictional force, Ff of the plane on the load acts parallel to the surface, and is always in a direction opposite to the motion of the object. It is equal to the normal force multiplied by the coefficient of static friction μ between the two surfaces.
Using Newton's second law of motion the load will be stationary or in steady motion if the sum of the forces on it is zero. Since the direction of the frictional force is opposite for the case of uphill and downhill motion, these two cases must be considered separately:
• Uphill motion: The total force on the load is toward the uphill side, so the frictional force is directed down the plane, opposing the input force.
The mechanical advantage is \mathrm{MA} = \frac {F_w}{F_i} = \frac {\cos \phi} { \sin (\theta + \phi ) } \, where \phi = \tan^{-1} \mu \,. This is the condition for impending motion up the inclined plane. If the applied force Fi is greater than given by this equation, the load will move up the plane.
• Downhill motion: The total force on the load is toward the downhill side, so the frictional force is directed up the plane.
The mechanical advantage is \mathrm{MA} = \frac {F_w}{F_i} = \frac {\cos \phi} { \sin (\theta - \phi ) } \, This is the condition for impending motion down the plane; if the applied force Fi is less than given in this equation, the load will slide down the plane. There are three cases:
1. \theta < \phi\,: The mechanical advantage is negative. In the absence of applied force the load will remain motionless, and requires some negative (downhill) applied force to slide down.
2. \theta = \phi\,: The ' angle of repose'. The mechanical advantage is infinite. With no applied force, load will not slide, but the slightest negative (downhill) force will cause it to slide.
3. \theta > \phi\,: The mechanical advantage is positive. In the absence of applied force the load will slide down the plane, and requires some positive (uphill) force to hold it motionless