Problem — Sliding Down an Inclined Plane with Friction

Problem

A body slides down an inclined plane of angle $\theta$ in the presence of kinetic friction $\mu_d$. Writing Newton’s second law along the plane and perpendicular to it, show that the body’s acceleration is $a = g(\sin\theta - \mu_d\cos\theta)$. At what minimum angle does the body start to move from rest, in the presence of static friction $\mu_s$?

Solution

Step 1 — Choice of axes. It is convenient to choose inclined axes: the $x$ axis along the plane, oriented in the direction of descent, and the $y$ axis perpendicular to the plane, pointing outward. With this choice the acceleration has only the $x$ component.

Step 2 — Resolving the weight. The weight $mg$ is vertical. Relative to the inclined axes it resolves into:

• a component along the plane (driving): $mg\sin\theta$;
• a component perpendicular to the plane: $mg\cos\theta$.

Step 3 — Equation perpendicular to the plane. The body neither leaves nor sinks into the plane, so $a_y = 0$:

$N - mg\cos\theta = 0 \quad\Longrightarrow\quad N = mg\cos\theta$

Step 4 — Kinetic friction force. The body is sliding down, so kinetic friction is directed up along the plane (opposing the motion) and has magnitude

$f_d = \mu_d N = \mu_d\, mg\cos\theta$

Step 5 — Equation along the plane. Along $x$, driving force minus friction equals mass times acceleration:

$mg\sin\theta - \mu_d\, mg\cos\theta = m\,a$

Dividing by $m$, the mass cancels:

$\ev{a = g(\sin\theta - \mu_d\cos\theta)}$

as required.

Step 6 — Minimum angle for slipping. At rest, static friction acts, and it can rise up to $f_{s,\max} = \mu_s N = \mu_s\, mg\cos\theta$. The body starts to move when the driving component exceeds the maximum static friction:

$mg\sin\theta > \mu_s\, mg\cos\theta \quad\Longrightarrow\quad \tan\theta > \mu_s$

The minimum angle is therefore the one for which $\tan\theta = \mu_s$:

$\ev{\theta_{\min} = \arctan\mu_s}$

Links

Topics: Friction Concepts: Newton’s second law · Kinetic friction Skills: Resolving forces on an inclined plane Objects: Inclined plane