Inclined plane
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 meters by 37.50 meters, which provides a mechanical advantage of

so a 100 lb tension force on the cable will lift a 4810 lb load. The grade of this incline is 2%, which means the angle θ is small enough that sin θ≈tan θ.
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