Question

Suppose you are standing on a train accelerating at 0.20g. What minimum coefficient of static friction must exist between your feet and the floor if you are not to slide?

Short answer

The coefficient of static friction is 0.20 between your feet and the floor.

Step-by-step solution

Step 1. Given data and assumption

The non-sliding condition is the static friction force, which is equal to the reaction force due to acceleration.

Given data:

The acceleration of the train is $a=0.20g$.

Assumption:

Let ${\mu }_{\mathrm{s}}$be the coefficient of static friction between your feet and the floor.

Step 2. Calculation of the coefficient of static friction

The normal force acting on you is $N=mg$.

Then, the static friction force is

$\begin{array}{c}{f}_s={\mu }_{\mathrm{s}}N\\ ={\mu }_{\mathrm{s}}mg\end{array}$

You do not slide, which means that the static friction force equals the force due to the train’s acceleration. So,

$\begin{array}{c}{f}_s=ma\\ {\mu }_{\mathrm{s}}mg=ma\\ {\mu }_{\mathrm{s}}=\frac{a}{g}\end{array}$

Now, substituting the values in the above equation,

$\begin{array}{c}{\mu }_{\mathrm{s}}=\frac{0.20g}{g}\\ =0.20\end{array}$

Hence, the coefficient of static friction is 0.20.