Question

A coffee cup on the horizontal dashboard of a car slides forward when the driver decelerates from 45 km/h to rest in 3.5 s or less, but not if she decelerates in a longer time. What is the coefficient of static friction between the cup and the dashboard? Assume that the road and the dashboard are level (horizontal).

Short answer

The coefficient of static friction between the cup and the dashboard is $0.36$.

Step-by-step solution

Step 1. Variables on which the frictional force depends

The frictional force's value can be evaluated by examining the frictional coefficient between the surface and the object and the normal force acting on the object.

Step 2. Given information

Given data:

The initial velocity is$u=45\mathrm{km}/\mathrm{h}$.

The final velocity is $t=3.5\mathrm{s}$.

The time is $t=3.5\mathrm{s}$.

Step 3. Calculate the acceleration of the system

The acceleration of the system can be calculated as

$\begin{array}{l}a=\frac{v-u}{t}\\ a=\frac{\left(0\right)-\left(45\mathrm{km}/\mathrm{h}\right)\left(\frac{1000\mathrm{m}}{1\mathrm{km}}\right)\left(\frac{1\mathrm{h}}{3600\mathrm{s}}\right)}{\left(3.5\mathrm{s}\right)}\\ a=-3.57\mathrm{m}{/\mathrm{s}}^2\end{array}$

Here, a negative sign indicates that the vehicle is decelerating.

Step 4. Calculate the coefficient of static friction

The frictional force, which balances the force acting on the coffee cup along with the motion, is

$\begin{array}{c}-F_r=ma\\ -\mu mg=ma\\ \mu =-\frac{a}{g}\end{array}$

Substituting the values in the above equation,

$\begin{array}{l}\mu =-\frac{\left(-3.57\mathrm{m}{/\mathrm{s}}^2\right)}{\left(9.8\mathrm{m}{/\mathrm{s}}^2\right)}\\ \mu =0.36\end{array}$

Thus, the coefficient of static friction is $0.36$.