8.15 Incline plane

Incline is a plane whose two ends are at different elevation with respect to a base line. This is a useful device to reduce the requirement of force in exchange of extended motion. When we push an object over an incline to raise the object at higher level, we need to apply lesser force (mg sinθ) in comparison to lifting the same vertically against its weight (mg).

What we gain through reduction in force we have to work longer to cover a longer length. Moreover, no plane, however smooth, is friction-less. Such devices aiming to reduce the magnitude of effort are ultimately less energy efficient.

Incline

Shape

An incline can have different shapes. Three general shapes, including single and double inclines, are shown here for illustration.

Forces

The force analysis with respect to motion of the block on a rough incline plane involves various forces. A minimum of three forces operate on the block placed on the incline : (i) weight of the block, mg, (ii) normal force (N) and (iii) friction ( $f_s$ , $F_s$ or $F_k$ ). Any other external force is additional to these three named forces. On the other hand, named forces reduce to two (weight and normal force), if the friction between incline and block is negligible to be considered zero.

Choice of planar coordinates

There are two useful coordinate orientations to analyze the forces and consequently the motion of the block.

We can either keep x-axis along incline and y-axis perpendicular to incline. In this setup, the weight of the block lying on the incline surface is resolved into components along these directions. The angle between vertical and perpendicular to incline is equal to the angle of incline “θ” as shown in the figure. The normal force on the block due to incline surface is in y-direction, whereas friction is along x-direction.

In the nutshell, contact forces are along the chosen coordinates, but weight of the body makes an angle with the axis, which needs to be resolved along the axes of the coordinate system for the analysis.

Alternatively, we can align x and y – axes along horizontal and vertical directions. In that case, weight of the block is along y-axis and as such need not be resolved. However, the two contact forces (normal and friction) on the block are now at an angle with the axes and are required to be resolved for force analysis.

It is evident, therefore, that the first choice is relatively better in most of the cases as there is only one force to be resolved into components against resolution of two forces as required in the second case.

Friction

The incline and block interface may be either termed as “smooth” or “rough”. The smooth surface indicates that we can neglect friction force. We should be aware that there are actually two contact interfaces as shown in the figure.