Chapter 5, Problem 9P

To determine

The maximum speed with which a $\text{1200}\text{kg}$ car can round a turn of the radius $90.0\text{m}$ on a flat road if the coefficient of friction between tires and road is $0.65$ .

Whether this result is independent of the mass of car.

Answer to Problem 9P

Solution:

When the car moves on a plane curve road, then static friction force providesthe necessary centripetal force. The maximumspeed of the car $\text{23}\text{.94}\text{m/s}$ .To avoid the skidding,the speed of car must be equal to $\text{23}\text{.94}\text{m/s}$

Explanation of Solution

Given:

Mass of car $\text{=1200}\text{kg}$

The radius of round turn $\text{=90}\text{.0}\text{m}$

Co-efficient of static friction $\left(\text{μs}\right)\text{=0}\text{.65}$

Formula used:

Static friction force:

  $f_s\le {\mu }_{s}N$

  

From Newton’s second law of motion:

  $\begin{array}{l}{\displaystyle \sum F_y=m{a}_y}\\ {\displaystyle \sum F_x=m{a}_x}\end{array}$

Calculation:

Apply Newton’s law in the vertical direction

  $\begin{array}{l}{\displaystyle \sum F_y=0}\\ N-mg=0\\ N=mg\end{array}$

Centripetal acceleration towards the center of the curve $=\frac{v^2}{r}$

This acceleration is provided by the frictionforce

So

  $\begin{array}{l}{f}_s=m\left(\frac{v^2}{r}\right)\\ f_s\le {\mu }_{s}N\\ m\left(\frac{v^2}{r}\right)\le {\mu }_{s}N\\ m\left(\frac{v^2}{r}\right)\le {\mu }_{s}mg\left(\because N=mg\right)\\ v\le \sqrt{{\mu }_{s}rg}\end{array}$

Yes,the result is independent of the mass of the car, its dependence upon static friction coefficient $\left({\text{μ}}_{\text{s}}\right)\text{,}$ radius of the curve, and acceleration due to gravity.

  $\begin{array}{l}v\le \sqrt{\left(0.65\right)\times \left(90.0\right)\times \left(9.80\right)}\\ v\le 23.94\text{m/s}\end{array}$

So, maximum speed of the particle is equal to $\text{23}\text{.94 m/s}$ , if speed is greater than this value, then the car will skid outside.

Conclusion:

When a car moves on a level curve road, the static friction provides necessary centripetal force, which keeps the car on the circular path. Static friction has a maximum limit, so speed also has a maximum limit, and the speed should be less than equal to that value.