Inclined plane
Frictionless inclined plane

If there is no friction between the object being moved and the plane, the device is called an ideal inclined plane. This condition might be approached if the object is rolling like a barrel, or supported on wheels or casters. Due to conservation of energy, for a frictionless inclined plane the work done on the load lifting it, W out {\displaystyle W_{\text{out}}}, is equal to the work done by the input force, W in {\displaystyle W_{\text{in}}}

Work is defined as the force multiplied by the displacement an object moves. The work done on the load is equal to its weight multiplied by the vertical displacement it rises, which is the "rise" of the inclined plane

The input work is equal to the force F i {\displaystyle F_{\text{i}}} on the object times the diagonal length of the inclined plane.

Substituting these values into the conservation of energy equation above and rearranging

To express the mechanical advantage by the angle θ {\displaystyle \theta } of the plane, it can be seen from the diagram (above) that

So

So the mechanical advantage of a frictionless inclined plane is equal to the reciprocal of the sine of the slope angle. The input force F i {\displaystyle F_{\rm {i}}} from this equation is the force needed to hold the load motionless on the inclined plane, or push it up at a constant velocity. If the input force is greater than this, the load will accelerate up the plane. If the force is less, it will accelerate down the plane.
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