Choice of axes

When a body sits on a plane inclined at angle $\theta$, the most convenient choice is not the usual horizontal/vertical axes, but axes rotated together with the plane:

• $x$ axis: along the plane, positive upwards (or downwards, as preferred);
• $y$ axis: perpendicular to the plane, positive outwards from the surface.

The reason is purely practical. With “tilted” axes the constraint reaction $\vec{R}$ has only one component (entirely along $y$) and the acceleration too has only one component (entirely along $x$, because the body does not leave the plane). The only force that needs to be resolved is the weight, and the working simplifies enormously.

Inclined plane: the $x$ and $y$ axes are rotated along the plane. The weight $\vec{P}$ is resolved into a component parallel to the plane ($mg\sin\theta$, which causes the motion) and one perpendicular to it ($mg\cos\theta$, balanced by $\vec{R}$).

In this system the components of the weight become (with $x$ positive upwards):

$\vec{P} = \begin{pmatrix} -mg\sin\theta \\ -mg\cos\theta \end{pmatrix}$

while the constraint reaction and the acceleration each have only one component:

$\vec{R} = \begin{pmatrix} 0 \\ R \end{pmatrix}, \qquad \vec{a} = \begin{pmatrix} a \\ 0 \end{pmatrix}$

Why it pays to rotate the axes

With rotated axes, Newton’s second law splits into two decoupled equations: one along the plane (giving the acceleration) and one perpendicular to it (giving the constraint reaction). No “mixed” components ever appear.

Links

Topics: Dinamica Concepts: Forza peso · Forza normale Skills: Scelta degli assi · Scomposizione sul piano inclinato · Diagramma di corpo libero Methods: Scomposizione in componenti cartesiane Objects: Piano inclinato

Related exercises: Mass on a smooth plane · Bank angle of a curve · Acceleration on an inclined plane with friction