Problem 101 - University Physics with Modern Physics

Question

A curve with a 120 -m radius on a level road is banked at the correct angle for a speed of 20 \(\mathrm{m} / \mathrm{s}\) . If an automobile rounds this curve at \(30 \mathrm{m} / \mathrm{s},\) what is the minimum coefficient of static friction needed between tires and road to prevent skidding?

Step-by-Step Solution

Answer

The minimum coefficient of static friction required is approximately 0.22.

Step 1: Understand the Problem

We have a curve of radius 120 m, banked for a speed of 20 m/s. We need to find the coefficient of static friction required for a car traveling at 30 m/s to prevent skidding.

Step 2: Find the Banking Angle

The banking angle is calculated using the formula: \( \tan \theta = \frac{v^2}{rg} \), where \( v \) is the speed for which the road is banked, \( r \) is the radius, and \( g \) is the acceleration due to gravity (approximately 9.8 m/s²). Substitute \( v = 20 \) m/s, \( r = 120 \) m, and \( g = 9.8 \) m/s² into the formula to compute \( \theta \).

Step 3: Calculate Banking Angle

Substitute the values into the formula: \( \tan \theta = \frac{(20)^2}{120 \times 9.8} = \frac{400}{1176} \approx 0.3401 \). The angle can be found using \( \theta = \tan^{-1}(0.3401) \approx 18.8^\circ \).

Step 4: Set Up the Forces for the New Speed

When the car travels at 30 m/s, both the friction and the banking angle contribute to providing the centripetal force. The formula is: \( v^2/r = g (\tan \theta + \mu) / (1 - \mu \tan \theta) \) where \( \mu \) is the friction coefficient, which we need to find.

Step 5: Plug Values into the Forces Equation

We already know \( \theta = 18.8^\circ \), \( v = 30 \) m/s, \( r = 120 \) m: \[ \frac{30^2}{120} = 9.8 \left( \tan(18.8^\circ) + \mu \right) / \left( 1 - \mu \tan(18.8^\circ) \right) \]. Solve for \( \mu \).

Step 6: Solve for \( \mu \)

Calculate the left-hand side: \( \frac{30^2}{120} = 7.5 \). Using \( \tan(18.8^\circ) \approx 0.3401 \), the equation becomes: \[ 7.5 = 9.8 \left( 0.3401 + \mu \right) / \left( 1 - 0.3401 \mu \right) \]. Solve this equation for \( \mu \).

Step 7: Simplification and Calculation

Rearranging and simplifying, we get: \[ 7.5 (1 - 0.3401 \mu) = 9.8 (0.3401 + \mu) \]. Expand and solve the equation for \( \mu \). After simplifying, find \( \mu \approx 0.22 \).

Key Concepts

Static Friction

Static friction is the frictional force that resists the initial movement of two surfaces in contact. It acts in the opposite direction to the applied force until the necessary force to start the movement is overcome. In our problem, static friction plays a vital role in preventing the car from skidding while negotiating the banked curve at a higher speed.

Static friction relies on the nature of the surfaces; in this case, the tires and the road. The maximum static frictional force one can exert is given by:

• \( f_{s_{max}} = \mu_s N \)

where \( \mu_s \) is the coefficient of static friction and \( N \) is the normal force. For a car moving along a banked curve, this static friction provides the necessary component to contribute to centripetal force when speed exceeds the intended design limit of the curve.

Understanding static friction helps identify how much grip the tires need on a surface to maintain control without slipping. As demonstrated in the exercise, knowing this coefficient is crucial for determining the threshold speed for safe travel on curved paths.​

Banked Curves

Banked curves are stretches of road that are tilted at an angle relative to the horizontal. This design allows vehicles to make turns at higher speeds than they would otherwise be capable of on flat surfaces by utilizing both gravitational force and the normal force to provide the required centripetal force.

When a road is properly banked, the force of gravity has a component directed towards the center of the curve, which assists in maintaining the vehicle's path. The banking angle \( \theta \) is crucial, as it determines the ideal speed \( v \) for which no friction is needed:

• \( \tan \theta = \frac{v^2}{rg} \)

where \( r \) is the curve radius and \( g \) is the acceleration due to gravity.

Understanding the role of a banked curve helps design roads that minimize reliance on friction, enabling safer conditions for vehicles traveling at a designated speed. In cases where speed exceeds the design speed, static friction becomes a critical factor as showcased in our exercise.

Centripetal Force

Centripetal force is the inward force required to keep an object moving in a circular path. In the context of vehicles rounding a banked curve, this force is necessary to alter the direction of the car's motion continuously, keeping it on the curved path rather than moving in a straight line.

The formula for centripetal force \( F_c \) required to maintain an object moving with speed \( v \) on a circular path of radius \( r \) is:

• \( F_c = \frac{mv^2}{r} \)

where \( m \) is the mass of the object. In a banked curve scenario, the centripetal force can be supplied by a combination of the normal force, gravitational force, and friction, depending on whether the vehicle is moving at the banked design speed, above, or below it.

A better grip on centripetal force assists in understanding why skidding occurs when the static friction is not enough to supply the extra force needed when vehicles exceed safe speed thresholds on curves.

Physics Equations

Physics equations are the bedrock of problem-solving in various real-world scenarios, including those involving motion on banked curves. These equations connect different physical quantities such as forces, motion, and mechanical energy, creating a framework that allows us to calculate unknown variables from known ones.

For the given exercise, several fundamental equations are in play:

• The equation for the banking angle involves the tangent function: \( \tan \theta = \frac{v^2}{rg} \).
• Centripetal force is given by \( F_c = \frac{mv^2}{r} \).
• The relation involving static friction and its role when speed is above the designed speed: \[ \frac{v^2}{r} = g (\tan \theta + \mu) / (1 - \mu \tan \theta) \]

These equations help us solve for the coefficient of static friction, emphasizing their importance in tackling complex physics problems steadily. By systematically applying these equations, students can dissect problems, gaining more profound insights and enhancing their analytical skills in physics.