Question

Question: Before 1960, people believed that the maximum attainable coefficient of static friction for an automobile tire on a roadway was ${\mathit{\mu }}_{\mathbf{s}}\mathbf{=}\mathbf{1}$. Around 1962, three companies independently developed racing tires with coefficients of $\mathbf{1}\mathbf{.}\mathbf{6}$. This problem shows that tires have improved further since then. The shortest time interval in which a piston-engine car initially at rest has covered a distance of one-quarter mile is about $\mathbf{4}\mathbf{.}\mathbf{43}\mathbf{}\mathbf{s}$. (a) Assume the car’s rear wheels lift the front wheels off the pavement as shown in Figure P5.58. What minimum value of ${\mathbf{\mu }}_{\mathbf{s}}$is necessary to achieve the record time? (b) Suppose the driver were able to increase his or her engine power, keeping other things equal. How would this change affect the elapsed time?

Short answer

(a) The minimum value of the coefficient of static friction is $4.18$.

(b) The time would decrease and car might flip.

Step-by-step solution

Write the given data from the question

Consider the given data as below.

The shortest time interval, $t=4.43\mathrm{s}$.

Covered distance, $s=\frac{1}{4}\mathrm{mile}=\frac{1}{4}\times 1609\mathrm{m}$.

Determine the required formulas

The expression to calculate the force on the object is given as follows:

$\mathit{F}\mathbf{=}\mathit{m}\mathit{a}$ ….. (i)

Here, $\mathit{F}$is the working acting on object, $\mathit{m}$is the mass of the object and $\mathit{a}$is the acceleration of the object.

The expression for the motion of equation is given as follows:

$\mathit{s}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathit{a}{\mathit{t}}^{\mathbf{2}}$ …… (ii)

Here, $\mathit{t}$is time and $\mathit{s}$is the displacement.

The expression for the static friction is given as follows:

$\mathit{F}\mathbf{=}{\mathit{\mu }}_{\mathbf{s}}\mathit{m}\mathit{g}$ ….. (iii)

Here, ${\mathit{\mu }}_{\mathbf{s}}$is the coefficient of static friction and $\mathit{g}$is the acceleration due to gravity.

 Step 3: (a) Calculate the minimum value of coefficient of static friction

Calculate the acceleration by substitute $4.43\mathrm{s}$for $t$ and $\frac{1}{4}\mathrm{mile}$for $s$into equation (ii).

role="math" localid="1663648676415" $$\begin{gathered} \frac{1}{4}\times 1609=\frac{1}{2}\times \mathrm{a}\times {\left(4.43\right)}^2 \\ \mathrm{a}=\frac{2}{4}\times \frac{1609}{{\left(4.43\right)}^2} \\ \mathrm{a}=40.99\mathrm{m}/{\mathrm{s}}^2 \end{gathered}$$

Derive the expression for the coefficient of static friction.

Substitute ${\mu }_{s}mg$for $F$into equation.

role="math" localid="1663648949654" $$\begin{gathered} {\mu }_{s}mg=ma \\ {\mu }_{s}g=a \\ {\mu }_s=\frac{a}{g}........\left(\mathrm{iv}\right) \end{gathered}$$

Calculate the minimum value of the coefficient of static friction.

Substitute $40.99\mathrm{m}/{\mathrm{s}}^2$for $a$ and $9.8\mathrm{m}/{\mathrm{s}}^2$for $g$into equation (iv).

${\mu }_s=\frac{40.99\mathrm{m}/{\mathrm{s}}^2}{9.81\mathrm{m}/{\mathrm{s}}^2}=4.18$

Hence, the minimum value of the coefficient of static friction is $4.18$.

(b) Determine the effect of changing power on elapse time

If the car engine's power increases and other values are the same, the car's wheel might skid, and only kinetic friction acts; eventually, the car might flip.

Let assume the power of the engine is double, therefore the acceleration is given by

$a=2{\mu }_{s}g$

Substitute $4.18$for ${\mu }_s$and $9.8\mathrm{m}/{\mathrm{s}}^2$for $g$into the above equation.

$\begin{array}{rcl}\mathrm{a}& =& 2\times 4.18\times 9.81\mathrm{m}/{\mathrm{s}}^2\\ & =& 82.0116\mathrm{m}/{\mathrm{s}}^2\end{array}$

Calculate the time that is required to cover the distance by substituting the known values into equation (ii).

$$\begin{gathered} \frac{1}{4}\times 1609=\frac{1}{2}\times 82.0116\times {\mathrm{t}}^2 \\ {\mathrm{t}}^2=\frac{2\times 1609}{4\times 82.0116} \\ \mathrm{t}=\sqrt{9.809}\mathrm{s} \\ \mathrm{t}=3.132\mathrm{s} \end{gathered}$$

From the above discussion, it can be seen that if the power is increased, the time is decreased, which causes the wheel of the car to skid, and the car might flip.