Problem 8 - College Physics

Question

A highway curve with radius 900.0 \(\mathrm{ft}\) is to be banked so that a car traveling 55.0 mph will not skid sideways even in the absence of friction. (a) Make a free-body diagram of this car. (b) At what angle should the curve be banked?

Step-by-Step Solution

Answer

The curve should be banked at an angle of approximately 12.77 degrees.

Step 1: Understanding the Problem

We need to determine the angle at which a highway curve should be banked so that a car moving at a certain speed does not skid sideways, even when friction is absent. The given values are the speed of the car, 55.0 mph, and the radius of the curve, 900.0 ft.

Step 2: Convert Units

Convert the car's speed from mph (miles per hour) to feet per second (ft/s) for consistency in our calculations. 1 mile = 5280 ft and 1 hour = 3600 seconds.So, 55 mph = \( 55 \times \frac{5280}{3600} \) = 80.67 ft/s.

Step 3: Identify the Forces

In the free-body diagram, consider the gravitational force acting downwards ( W = mg ), the normal force perpendicular to the road surface ( N ), and the centripetal force needed to keep the car moving in a circle.

Step 4: Relate Forces to Banking Angle

On a banked curve, the normal force provides the centripetal force required to keep the car in circular motion. Set the centripetal force equal to the horizontal component of the normal force:\( N \, \sin\theta = \frac{mv^2}{r} \).Additionally, the vertical component of the normal force balances the gravitational force:\( N \, \cos\theta = mg \).

Step 5: Solve for the Banking Angle

From the two force equations: \( \frac{\sin\theta}{\cos\theta} = \frac{v^2}{rg} \) which simplifies to \( \tan\theta = \frac{v^2}{rg} \).Calculate \( \theta \) using: \( \tan\theta = \frac{(80.67)^2}{(900)(32.2)} \).

Step 6: Calculate the Angle

Substitute the numbers:\( \tan\theta = \frac{6508.45}{28980} = 0.2247 \).Thus, \( \theta = \tan^{-1}(0.2247) = 12.77^\circ \).

Key Concepts

Centripetal Force

Centripetal force is essential for keeping an object moving in a circular path. Imagine a car driving around a curve on the road. The centripetal force is what prevents it from skidding outwards due to its inertia. This force is directed towards the center of the circle, ensuring that the car maintains its circular motion.

The formula for calculating centripetal force is:

• \( F_c = \frac{mv^2}{r} \)

where \( m \) is the mass of the car, \( v \) is its velocity, and \( r \) is the radius of the curve.

In this exercise, a banked curve is designed to provide centripetal force without relying on friction. Understanding how centripetal force works helps in setting up the necessary conditions for safe and efficient turning on curves.

Banked Curve

A banked curve is a roadway or track that is raised on the outer edge compared to the inner edge. The purpose of this design is to allow vehicles to navigate the curve at higher speeds without the risk of skidding, even in the absence of friction.

When a car takes a banked curve, the design of the road provides the necessary centripetal force through the component of the normal force. This ensures that the car stays on its path despite the curve. The banking angle plays a critical role in achieving this by creating an appropriate path for the force vector.

• In a banked curve, the horizontal component of the normal force acts as the centripetal force.
• The angle of banking reduces the reliance on friction.

Banked curves are common in highways and racing tracks, where maintaining speed and safety are priorities.

Free-Body Diagram

A free-body diagram is a visual representation that helps in isolating the forces acting on an object. In the scenario of a car navigating a banked curve, it's vital to understand these forces to determine how the car can take the turn without skidding.

Here are the main forces represented in the free-body diagram for a car on a banked curve:

• Gravitational force (\( mg \)), acting vertically downward.
• Normal force (\( N \)), acting perpendicular to the surface of the road.
• Centripetal force component, coming from the horizontal component of the normal force.

Drawing a free-body diagram aids in analyzing these forces and understanding how they interact to achieve equilibrium and maintain motion on the banked turn. This visualization helps to formulate the required equations for solving physics problems.

Angle Calculation

The calculation of the banking angle is key to ensuring vehicles can safely take a curve without relying on friction. The angle determines how much the road should be inclined so that the centripetal force required for navigating the turn comes completely from the road design and not from external factors like tire friction.

The relationship for calculating the banking angle \( \theta \) is:

• \( \tan\theta = \frac{v^2}{rg} \)

where \( v \) is the velocity of the vehicle, \( r \) is the radius of the curve, and \( g \) is the acceleration due to gravity.

In the given exercise, converting the vehicle's speed and using this formula allows the calculation of the exact angle. By substituting the appropriate values, we find that \( \theta = 12.77^\circ \). This calculation ensures the car can travel safely along the banked curve at the designated speed.