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 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: : 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. : 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. : 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 Mechanical advantage using power The mechanical advantage of an inclined plane is the ratio of the weight of the load on the ramp to the force required to pull it up the ramp. If energy is not dissipated or stored in the movement of the load, then this mechanical advantage can be computed from the dimensions of the ramp. In order to show this, let the position r of a rail car on along the ramp with an angle, θ, above the horizontal be given by where R is the distance along the ramp. The velocity of the car up the ramp is now Because there are no losses, the power used by force F to move the load up the ramp equals the power out, which is the vertical lift of the weight W of the load. The input power pulling the car up the ramp is given by and the power out is Equate the power in to the power out to obtain the mechanical advantage as The mechanical advantage of an inclined plane can also be calculated from the ratio of length of the ramp L to its height H, because the sine of the angle of the ramp is given by therefore, Example: If the height of a ramp is H = 1 meter and its length is L = 5 meters, then the mechanical advantage is which means that a 20 lb force will lift a 100 lb load. The Liverpool Minard inclined plane has the dimensions 1804