Problem 59 - Physics for Scientists and Engineers with Modern Physics

Question

(III) A curve of radius \(68 \mathrm{~m}\) is banked for a design speed of \(85 \mathrm{~km} / \mathrm{h}\). If the coefficient of static friction is 0.30 (wet pavement), at what range of speeds can a car safely make the curve? [Hint: Consider the direction of the friction force when the car goes too slow or too fast.]

Step-by-Step Solution

Answer

The safe speed range is approximately 78.08 km/h to 105.77 km/h.

Step 1: Convert Speed to Meters per Second

The given design speed is 85 km/h. First, convert this speed into meters per second using the conversion factor: 1 km/h = 0.27778 m/s.\[v = 85 \text{ km/h} \times 0.27778 \text{ m/s per km/h} = 23.61 \text{ m/s}\]

Step 2: Calculate Banking Angle

Use the formula for banking angle for the design speed:\[\tan(\theta) = \frac{v^2}{rg}\]Where \(v\) is the speed in m/s, \(r\) is the radius (68 m), and \(g\) is the acceleration due to gravity (9.8 m/s²):\[\tan(\theta) = \frac{(23.61)^2}{68 \times 9.8}\]Solve for \(\theta\):\[\tan(\theta) = 0.825\]\[\theta = \tan^{-1}(0.825) \approx 39.39^\circ\]

Step 3: Find Range of Speeds with Friction

Determine the range of speeds where friction helps the car stay on the banked curve:1. **Maximum Speed:**The maximum speed occurs when the frictional force is directed down the incline. Use the equation:\[v_{\text{max}} = \sqrt{\frac{rg(\tan(\theta) + \mu_s)}{1 - \mu_s \tan(\theta)}}\]Substitute \(\mu_s = 0.3\), \(r = 68\text{ m}\), and \(\theta = 39.39^\circ\):\[v_{\text{max}} = \sqrt{\frac{68 \times 9.8 (0.825 + 0.3)}{1 - 0.3 \times 0.825}} = \sqrt{86.244} \approx 29.385 \text{ m/s}\]2. **Minimum Speed:**The minimum speed occurs when the frictional force is directed up the incline:\[v_{\text{min}} = \sqrt{\frac{rg(\tan(\theta) - \mu_s)}{1 + \mu_s \tan(\theta)}}\]Substitute \(\mu_s = 0.3\), \(r = 68\text{ m}\), and \(\theta = 39.39^\circ\):\[v_{\text{min}} = \sqrt{\frac{68 \times 9.8 (0.825 - 0.3)}{1 + 0.3 \times 0.825}} = \sqrt{47.025} \approx 21.69 \text{ m/s}\]

Step 4: Conclusion

Convert the speeds from m/s back to km/h:- Minimum speed \(v_{\text{min}} = 21.69 \text{ m/s} \times 3.6 = 78.08 \text{ km/h}\)- Maximum speed \(v_{\text{max}} = 29.385 \text{ m/s} \times 3.6 = 105.77 \text{ km/h}\)Thus, the car can safely navigate the curve at speeds ranging from approximately 78.08 km/h to 105.77 km/h.

Key Concepts

Frictional Forces

Frictional forces play a crucial role in how vehicles maneuver through banked curves, especially under varying speed conditions. When a car navigates a curve, the friction between its tires and the road surface can act in different directions, depending on whether the car is traveling at a higher or lower speed than the design speed.
In the context of banked curves, frictional forces can either aid or oppose the centripetal force required for circular motion. At high speeds, friction operates downwards along the incline to help in centripetal acceleration, preventing the car from skidding outwards.
Conversely, when the car travels at a speed lower than the ideal speed for the curve, static friction directs upwards, working against gravity to help maintain the car's path without sliding down the incline. The coefficient of static friction, denoted as \(\mu_s\), quantifies this frictional capability, typical values on wet pavement can significantly influence the safe speed range.

Circular Motion

Understanding circular motion is central to solving problems involving banked curves. A banked curve is designed to permit vehicles to follow a circular path without relying entirely on friction. In the banking scenario, a part of the net force acting towards the circle's center is produced by the banking of the road itself.
When a car travels around a curve, it naturally undergoes centripetal acceleration, which is directed toward the curve's center. The formula used in circular motion to determine this force is: \[ F_c = \frac{mv^2}{r} \] where \

Physics Problem Solving

Physics problem solving requires a systematic approach when dealing with banked curves. These problems usually require a mix of algebraic manipulation and understanding the physics concepts like forces and acceleration.
Approaching a banked curve problem starts with understanding the different forces acting on the vehicle. Consider gravitational force, normal force, and friction, all of which contribute differently based on the vehicle's speed and the curve's angle.

• First, transform all speeds into consistent units, such as meters per second, for accurate calculations.
• Utilize equations like \( \tan(\theta) = \frac{v^2}{rg} \) to find the banking angle \( \theta \).
• Analyze maximum and minimum speeds by solving quadratic relationships, considering the static friction coefficient.

Physics problems in this context not only exercise mathematical proficiency but also demand a thorough comprehension of dynamics principles.

Banking Angle Calculations

Banking angle calculations are critical in determining how effectively a car can navigate a curve at various speeds. The angle ensures that a part of the gravitational force contributes to the centripetal force, reducing reliance on friction alone.
The tangent of the banking angle \( \theta \) is related to the design speed through the equation: \[ \tan(\theta) = \frac{v^2}{rg} \] Here, \( v \) is the design speed, \( r \) is the curve radius, and \( g \) is the acceleration due to gravity. Solving this gives the ideal angle, ensuring adequate centripetal force.
Calculating this angle accurately enables engineers to design safe roadways. It ensures that vehicles maintaining the design speed require very minimal frictional force to stay on course. Hence, it identifies the safe operational speed range and further aids in defining speed limits for safety purposes.