(II) How large must the coefficient of static friction be between the tires and the road if a car is to round a level curve of radius 85 m at a speed of 95 km/h?

Verified step by step guidance

1. Convert the speed of the car from km/h to m/s by using the conversion factor 1 km/h = 0.27778 m/s.
2. Use the formula for centripetal force, which is required to keep the car moving in a circular path: \(F_c = \frac{mv^2}{r}\), where \(m\) is the mass of the car, \(v\) is the speed of the car in m/s, and \(r\) is the radius of the curve.
3. Recognize that the centripetal force is provided by the frictional force between the tires and the road. The maximum frictional force that can act without slipping is given by \(F_f = \mu_s N\), where \(\mu_s\) is the coefficient of static friction and \(N\) is the normal force.
4. Since the car is on a level curve, the normal force \(N\) is equal to the gravitational force, which is \(mg\), where \(g\) is the acceleration due to gravity (approximately 9.8 m/s^2).
5. Set up the equation \(\frac{mv^2}{r} = \mu_s mg\) and solve for \(\mu_s\) by simplifying the equation to \(\mu_s = \frac{v^2}{rg}\). Substitute the values of \(v\), \(r\), and \(g\) to find the required coefficient of static friction.

Key Concepts

Coefficient of Static Friction

The coefficient of static friction is a dimensionless value that represents the ratio of the maximum static frictional force between two surfaces to the normal force pressing them together. It determines how much force is needed to overcome the initial resistance to motion. In the context of a car rounding a curve, a higher coefficient indicates better grip between the tires and the road, allowing the car to maintain its circular path without slipping.

Centripetal Force

Centripetal force is the net force required to keep an object moving in a circular path, directed towards the center of the circle. For a car rounding a curve, this force is provided by the friction between the tires and the road. The necessary centripetal force can be calculated using the formula F_c = (mv^2)/r, where m is the mass of the car, v is its speed, and r is the radius of the curve.

Kinematics and Speed Conversion

Kinematics is the branch of physics that deals with the motion of objects. In this problem, it is essential to convert the car's speed from kilometers per hour to meters per second to ensure consistency in units when applying formulas. The conversion is done by multiplying the speed in km/h by (1000 m/1 km) and dividing by (3600 s/1 h), resulting in a speed that can be directly used in calculations involving distance and time.