Editing Inclined plane (section)

==Mechanical advantage using power==
[[File:Free body.svg|thumb|Key: N = [[Normal force]] that is perpendicular to the plane, W=mg, where m = [[mass]], g = [[gravity]], and θ ([[theta]]) = Angle of inclination of the plane]]
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
:<math>\mathbf{r} = R (\cos\theta, \sin\theta),</math>

where ''R'' is the distance along the ramp.  The velocity of the car up the ramp is now

:<math>\mathbf{v} = V (\cos\theta, \sin\theta).</math>

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

:<math>P_{\mathrm{in}} = F V,\!</math>

and the power out is

:<math>P_{\mathrm{out}} = \mathbf{W}\cdot\mathbf{v} = (0, W)\cdot V (\cos\theta, \sin\theta) = WV\sin\theta.</math>

Equate the power in to the power out to obtain the mechanical advantage as

:<math> \mathrm{MA} = \frac{W}{F} = \frac{1}{\sin\theta}.</math>

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
:<math> \sin\theta = \frac{H}{L},</math>
therefore,
:<math> \mathrm{MA} = \frac{W}{F} = \frac{L}{H}.</math>

[[File:Plan incliné machine stationnaire Liverpool Minard.jpg|upright=1.3|thumb|Layout of the cable drive system for the Liverpool Minard inclined plane.]]

Example: If the height of a ramp is H = 1 meter and its length is L = 5 meters, then the mechanical advantage is
:<math> \mathrm{MA} = \frac{W}{F} = 5,</math>
which means that a 20&nbsp;lb force will lift a 100&nbsp;lb load.

The Liverpool Minard inclined plane has the dimensions 1804 meters by 37.50 meters, which provides a mechanical advantage of
:<math> \mathrm{MA} = \frac{W}{F} = 1804/37.50 = 48.1,</math>
so a 100&nbsp;lb tension force on the cable will lift a 4810&nbsp;lb load.  The grade of this incline is 2%, which means the angle θ is small enough that sin&nbsp;θ≈tan&nbsp;θ.