Answer to Question #104306 in Classical Mechanics for Baldo

Question

1. A corner in a flat road has a constant radius of r = 25 m. Air resistance and rolling resistance are zero. And the car is traveling at constant speed. What is the maximum speed our car can go round this corner without skidding? For a reasonably clean, dry road, take the coefficients of static and kinetic friction to be μs = 1.0 and μk=0.80. Hint: What horizontal force is acting on the car? Maximum speed _____ kph

2. You place a mass m = 25 on a book and slowly increase the angle θ that the book makes with the horizontal. At a critical angle θc = 43∘, the mass starts to slide so you hold the angle constant. The mass takes 0.75 s to slide a distance of 17 cm to the edge of the book. Assume that the values of μs are uniform all over the book. What is the value of μs ?

3. Using the information from #2 : What is the value of μk ?

Answer 1

1. A turning car experiences a centripetal force that makes the car change its direction of motion. This force is expressed as

$$F=m\frac{v^2}{R}.$$

According to Newton's second law, the car does not go skidding if this force is equal to the force of static friction:

$$m\frac{v^2}{R}=\mu_s mg,\\ \space\\ v=\sqrt{\mu_s gR}=\sqrt{1\cdot9.8\cdot25}=15.7\text{ m/s}.$$

To convert it to km/h, multiply by 3600 and divide by 1000:

$$v=15.7\cdot3600/1000=56.5\text{ km/h}.$$

2. No matter how the book slides, the coefficient of static friction can be easily found by calculating the tangent of the angle at which the body starts sliding:

$$\mu_s=\text{tan}\theta=\text{tan}43^\circ=0.93.$$

3. Draw the situation:

Calculate the acceleration:

$$a=\frac{2x}{t^2}.$$

According to Newton's second law:

$$N-mg\text{ cos}\theta=0,\\ -\mu_k N+mg\text{ sin}\theta=ma,\\ \mu_k=\text{tan}\theta-\frac{a}{g\text{ cos}\theta}=\\ \space\\=\text{tan}\theta-\frac{2x}{gt^2\text{ cos}\theta}=0.85.$$