up the plane. The mechanical advantage is The MA of an ideal inclined plane without friction is sometimes called ideal mechanical advantage (IMA) while the MA when friction is included is called the actual mechanical advantage (AMA). 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, Wout, is equal to the work done by the input force, Win 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 Fi 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 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 Fi 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. 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, Wfric, 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 Ff opposing it. The maximum friction force is given by where