Worked exercise: kinetic friction from a kinematic measurement

Problem

A block is released from rest at the top of an inclined plane of $\theta = 30^\circ$ and travels $s = 2\ \mathrm{m}$ in $t = 1{,}5\ \mathrm{s}$. Determine the coefficient of kinetic friction $\mu_d$.

Block released from rest: from the distance $s$ covered in time $t$ we work back to the acceleration and hence to the friction.

Solution

Method. The motion is uniformly accelerated from rest: $s = \tfrac{1}{2}at^2$, from which: $a = \frac{2s}{t^2} = \frac{4}{2{,}25} \approx 1{,}78\ \mathrm{m/s}^2$

From the dynamics of the plane with friction, $a = g(\sin\theta - \mu_d\cos\theta)$, we obtain: $\mu_d = \frac{g\sin\theta - a}{g\cos\theta} = \frac{9{,}81\cdot 0{,}5 - 1{,}78}{9{,}81\cdot 0{,}866} \approx \frac{3{,}125}{8{,}495} \approx 0{,}368$

$\ev{\mu_d = \frac{g\sin\theta - a}{g\cos\theta} \approx 0{,}368}$

Kinematics thus becomes a scale for weighing hidden forces: a stopwatch and a tape measure are enough to measure friction.

Links

Topics: Dinamica · Attrito Concepts: Attrito dinamico · Moto uniformemente accelerato · Seconda legge di Newton Skills: Scomposizione sul piano inclinato Objects: Piano inclinato · Blocco