Worked exercise — block on an inclined plane with 3 forces

Problem

A block of $5\;\text{kg}$ on a plane inclined at $35^\circ$ is pushed by a force $\vv{P}_{\text{app}} = 45\;\text{N}$ at $25^\circ$ from the surface. Find the acceleration and the constraint reaction.

Solution

Circling the block in the interaction diagram, we count 3 wavy lines: the weight (Earth), the constraint reaction $\vv{R}$ (Plane) and the applied force $\vv{F}$ (Hand). It is convenient to choose axes rotated along the plane: $x$ upward along the surface, $y$ perpendicular and outward. The constraint reaction thus has only one component ($y$) and the acceleration only one ($x$, along the plane). In these axes the three forces and the acceleration have components

$\vv{P}=\begin{pmatrix} -mg\sin 35^\circ \\ -mg\cos 35^\circ \end{pmatrix},\quad \vv{F}=\begin{pmatrix} F\cos 25^\circ \\ F\sin 25^\circ \end{pmatrix},\quad \vv{R}=\begin{pmatrix} 0 \\ R \end{pmatrix},\quad \vv{a}=\begin{pmatrix} a \\ 0 \end{pmatrix}$

Newton’s second law $\vv{P} + \vv{F} + \vv{R} = m\vv{a}$ gives two scalar equations, one along the plane and one perpendicular to it:

$F\cos 25^\circ - mg\sin 35^\circ = m\,a \qquad\qquad R + F\sin 25^\circ - mg\cos 35^\circ = 0$

From the perpendicular equation we isolate the constraint reaction:

$R = mg\cos 35^\circ - F\sin 25^\circ = 40{,}1 - 19{,}0 = 21{,}1\;\text{N}$

and from the one along the plane we obtain the acceleration (positive: the block accelerates up the slope):

$a = \frac{F\cos 25^\circ - mg\sin 35^\circ}{m} = \frac{40{,}8 - 28{,}1}{5{,}0} = 2{,}5\;\text{m/s}^2$

$\ev{a \approx 2{,}5\;\text{m/s}^2}$

$\ev{R \approx 21\;\text{N}}$

In depth — rotating the axes

Whenever an inclined plane is present, it is convenient to rotate the axes along the plane: the constraint reaction has only one component (along $y$), the acceleration only one (along $x$), and the calculations simplify enormously compared with horizontal/vertical axes.

Links

Topics: Dynamics Concepts: Newton’s second law · Normal force Skills: Resolution on the inclined plane · Choice of axes Objects: Inclined plane · Block