Question

A 1125-kg car and a 2250-kg pickup truck approach a curve on a highway that has a radius of 225 m. (a) At what angle should the highway engineer bank this curve so that vehicles traveling at 65.0 mi/h can safely round it regardless of the condition of their tires? Should the heavy truck go slower than the lighter car? (b) As the car and truck round the curve at 65.0 mi/h, find the normal force on each one due to the highway surface.

Short answer

(a) The banking angle should be approximately \( 22.44^\circ \). The truck does not need to go slower than the car. (b) The normal forces are approximately 11980.9 N for the car and 23961.8 N for the truck.

Step-by-step solution

Convert Speed to Meters per Second

First, we need to convert the speed from miles per hour to meters per second since SI units are generally used in physics. Use the conversion factor: 1 mile = 1609.34 meters and 1 hour = 3600 seconds. Thus, \( 65.0 \, \text{mi/h} = 65.0 \times \frac{1609.34}{3600} \, \text{m/s} \approx 29.06 \, \text{m/s} \).

Determine the Banking Angle

To find the angle \( \theta \) at which to bank the curve so that a vehicle can make the curve without relying on friction, use the formula \( \tan(\theta) = \frac{v^2}{rg} \), where \( v = 29.06 \, \text{m/s} \), \( r = 225 \, \text{m} \), and \( g = 9.8 \, \text{m/s}^2 \). Solving for \( \theta \), we have: \( \theta = \tan^{-1}\left(\frac{(29.06)^2}{225 \times 9.8}\right) \approx 22.44^\circ \).

Conclusion on Speed Differences

The need for the banking angle equation shows that it is designed to account for vehicles of different masses traveling at the same speed. Therefore, the heavy truck does not need to go slower than the lighter car if the curve is properly banked.

Calculate Normal Force on Car

The normal force \( N \) for a vehicle on a banked curve is given by a rearrangement of the banking conditions. For the car: \( N = \frac{mg}{\cos(\theta)} \). Plug \( m = 1125 \, \text{kg}, g = 9.8 \, \text{m/s}^2, \theta = 22.44^\circ \): \( N = \frac{1125 \times 9.8}{\cos(22.44^\circ)} \approx 11980.9 \, \text{N} \).

Calculate Normal Force on Truck

Similarly, use the rearranged normal force formula for the truck: \( N = \frac{mg}{\cos(\theta)} \). For the truck: \( m = 2250 \, \text{kg} \), \( N = \frac{2250 \times 9.8}{\cos(22.44^\circ)} \approx 23961.8 \, \text{N} \).

Key concepts

Understanding Circular Motion

Circular motion occurs when an object moves along a circular path. This movement is characterized by a constant change in the direction of velocity while maintaining a consistent speed. In the context of vehicles navigating a curve, the inward force required to keep the vehicle on this path is called the centripetal force. This force ensures the vehicle does not veer off its intended path. In our exercise, when a car or truck approaches the highway curve, it experiences this type of motion. The centripetal force needed to keep the vehicle stable is provided by both the friction between the tires and the road and, primarily, by the banked angle of the curve. Incorporating these factors allows vehicles to maintain their paths safely, regardless of the road conditions.

Navigating Banked Curves

Banked curves are designed to help vehicles maintain traction and navigate curves safely even in adverse conditions. The road surface on such curves is tilted at an angle relative to the horizontal. This angle is known as the banking angle. The banking angle allows the normal force exerted by the road to contribute to the centripetal force. By doing so, it reduces the reliance on friction alone to keep the vehicle on the curve. From the provided solution, we calculate the banking angle using the formula \( \tan(\theta) = \frac{v^2}{rg} \). Here, \( v \) indicates velocity, \( r \) the radius of the curvature, and \( g \) the acceleration due to gravity. The calculations showed that a banking angle of approximately 22.44 degrees is optimal, ensuring any vehicle can safely pass through the curve at the regulated speed without the need for friction.

The Role of Normal Force

Normal force is essential in analyzing vehicle dynamics on banked curves. This force acts perpendicular to the surface of the road and supports the weight of the vehicle. It alters slightly depending on the banking angle, which helps maintain the required centripetal force. For vehicles on a banked curve, the normal force can be calculated using the equation \( N = \frac{mg}{\cos(\theta)} \). Here, \( m \) represents the vehicle's mass, \( g \) is the acceleration due to gravity, and \( \theta \) is the banking angle. Our solutions reveal that, for the car with a mass of 1125 kg and the banking angle of 22.44 degrees, the normal force is approximately 11981 N. Similarly, a larger normal force of about 23962 N is experienced by the 2250 kg truck, reflecting its greater mass.

Essentials of Vehicle Dynamics

Vehicle dynamics refers to how a vehicle moves in response to different forces. These include forces generated during acceleration, braking, and turning. It's a key aspect of automotive engineering, ensuring safety and efficiency. On curved paths, dynamics play a vital role in understanding how vehicles interact with the road. The study of vehicle dynamics includes analyzing how different factors, such as vehicle mass and speed, affect the movement of the vehicle. In the case of banked curves, dynamics explain why heavy trucks do not need to drive at different speeds than lighter cars on a properly banked curve. Despite their different masses, both vehicles can safely round the curve at the same speed due to the calculated banking angle, which balances the centripetal and gravitational forces involved in circular motion. With a well-engineered banked curve, vehicle dynamics show us that mass does not dictate speed, rather the design of the road does.