Smooth inclined plane

On the frictionless inclined plane only two forces act: the weight $\vec{P}$ and the constraint reaction $\vec{R}$. Resolving along the rotated axes (see Choice of axes), Newton’s second law $\vec{P} + \vec{R} = m\vec{a}$ splits into two independent equations:

$\begin{cases} -mg\sin\theta = ma \\ -mg\cos\theta + R = 0 \end{cases}$

The first equation gives the acceleration along the plane, the second the constraint reaction:

Key formula — Smooth inclined plane

$a = g\sin\theta \qquad R = mg\cos\theta$

Two important physical observations. First: the acceleration does not depend on the mass. All bodies, heavy or light, slide down a smooth plane with the same acceleration $g\sin\theta$ — exactly as in free fall, of which the inclined plane is a “diluted” version (for $\theta = 90^\circ$ we recover $a = g$). Second: the constraint reaction $R = mg\cos\theta$ balances only the component of the weight pressing against the plane, and is therefore smaller than the total weight $mg$ (except when $\theta = 0$).

Limiting cases

As $\theta \to 0$ the plane becomes horizontal: $a \to 0$ and $R \to mg$ (the body stays still, the reaction carries the whole weight). As $\theta \to 90^\circ$ the plane becomes vertical: $a \to g$ (free fall) and $R \to 0$ (the plane no longer presses).

Links

Topics: Dinamica Concepts: Seconda legge di Newton · Forza peso · Forza normale Skills: Scomposizione sul piano inclinato · Applicazione delle leggi di Newton Objects: Piano inclinato · Blocco

Related exercises: Mass on a smooth plane · Acceleration on an inclined plane with friction · Spring on an inclined plane