Inclined plane
Inclined plane with friction

Where there is friction between the plane and the load, as for example with a heavy box being slid up a ramp, some of the work applied by the input force is dissipated as heat by friction, W fric {\displaystyle W_{\text{fric}}}, so less work is done on the load. Due to conservation of energy, the sum of the output work and the frictional energy losses is equal to the input work

Therefore, more input force is required, and the mechanical advantage is lower, than if friction were not present.With friction, the load will only move if the net force parallel to the surface is greater than the frictional force F f {\displaystyle F_{\text{f}}} opposing it. The maximum friction force is given by

where F n {\displaystyle F_{\text{n}}} is the normal force between the load and the plane, directed normal to the surface, and μ {\displaystyle \mu } is the coefficient of static friction between the two surfaces, which varies with the material. When no input force is applied, if the inclination angle θ {\displaystyle \theta } of the plane is less than some maximum value ϕ {\displaystyle \phi } the component of gravitational force parallel to the plane will be too small to overcome friction, and the load will remain motionless. This angle is called the angle of repose and depends on the composition of the surfaces, but is independent of the load weight. It is shown below that the tangent of the angle of repose ϕ {\displaystyle \phi } is equal to μ {\displaystyle \mu }

With friction, there is always some range of input force F i {\displaystyle F_{\text{i}}} for which the load is stationary, neither sliding up or down the plane, whereas with a frictionless inclined plane there is only one particular value of input force for which the load is stationary.
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