Problem 48 - College Physics

Question

A \(1125 \mathrm{~kg}\) car and a \(2250 \mathrm{~kg}\) pickup truck approach a curve on the expressway that has a radius of \(225 \mathrm{~m}\). (a) At what angle should the highway engineer bank this curve so that vehicles traveling at 65.0 mph 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 \(65.0 \mathrm{mph}\), find the normal force on each one due to the highway surface.

Step-by-Step Solution

Answer

The curve should be banked at an angle of approximately \(20.97^\circ\). Vehicle weight does not affect safe travel speed. The normal force is approximately \(1176.29 \text{ N}\) for the car and \(2352.58 \text{ N}\) for the truck.

Step 1: Understand the problem requirements

We need to calculate the banking angle required for vehicles to navigate a curve safely at a speed of 65.0 mph without relying on friction. Additionally, we must determine if vehicle weight affects speed requirements and calculate the normal force exerted on both vehicles.

Step 2: Convert speed from mph to m/s

Convert the speed of 65.0 mph to meters per second (m/s). Use the conversion factor: \[1 \text{ mph} = 0.44704 \text{ m/s}.\] Thus, \[65.0 \text{ mph} \times 0.44704 = 29.0576 \text{ m/s}.\]

Step 3: Apply laws of motion for banking angle calculation

The formula to find the banking angle \(\theta\) without relying on friction is: \[\tan \theta = \frac{v^2}{rg},\] where \(v\) is the velocity, \(r\) is the radius, and \(g\) is the acceleration due to gravity (\(9.81 \text{ m/s}^2\)).

Step 4: Calculate the required banking angle

Substitute values into the banking angle formula: \[\tan \theta = \frac{(29.0576)^2}{225 \times 9.81}.\] Calculate\[\tan \theta \approx \frac{844.347}{2207.25} \approx 0.3828.\] Thus, \(\theta \approx \tan^{-1}(0.3828) \approx 20.97^\circ\).

Step 5: Analyze vehicle weight impact on speed

Since the formula \(\tan \theta = \frac{v^2}{rg}\) does not include the mass of the vehicle, we conclude that vehicle weight does not affect the required speed for safe travel around the curve.

Step 6: Calculate the normal force for the car and truck

The normal force \(N\) is calculated using \(N = \frac{mg}{\cos \theta}\) due to the banking angle:- For the car: \(m = 1125 \text{ kg}\), so \[N_{car} = \frac{1125 \times 9.81}{\cos(20.97^\circ)} \approx 1176.29 \text{ N}.\]- For the truck: \(m = 2250 \text{ kg}\), so \[N_{truck} = \frac{2250 \times 9.81}{\cos(20.97^\circ)} \approx 2352.58 \text{ N}.\]

Key Concepts

Laws of Motion

Understanding the laws of motion is crucial when analyzing how vehicles interact with curves at certain speeds. Isaac Newton's laws of motion explain the relationships between a body and the forces acting upon it. More specifically, when a vehicle rounds a curve, it experiences centripetal force, which is necessary to keep it moving in a circular path rather than continuing straight. This force generates from the natural push of the pavement against the tires.
The first law of motion posits that a vehicle will maintain its state of rest or uniform motion unless acted upon by an external force. As vehicles enter a curve at 65.0 mph, centripetal force is required for it to change direction and follow the curve's path. This force is influenced by the speed (\( v \)), the mass of the vehicle, and the radius (\( r \)) of the curve. Understanding these basic principles allows us to grasp why banking angles are utilized in highway design. By angling the road, one can mitigate dependence on friction, thereby relying on gravity and the normal force to provide the centripetal force needed to navigate safely around a curve.

Normal Force Calculation

Normal force plays a pivotal role in the dynamics of vehicle movement on a banked curve. To comprehend how it functions, we need to delve into basic principles of physics governing forces. In this scenario, normal force is the perpendicular force exerted by a surface to support the weight of an object resting on it.For a vehicle on a banked curve, the normal force must be sufficient to counteract the downward force of gravity, while also contributing to the centripetal force needed to maintain the vehicle in curved motion. The formula to determine the normal force (\( N \)) when a banking angle(\( \theta \)) is involved is given by:

• \( N = \frac{mg}{\cos \theta} \)

where:

• \( m \) is the mass of the vehicle,
• \( g \) is the acceleration due to gravity, and
• \( \theta \) is the banking angle.

Calculating this force for both the car and the truck as they round the curve ensures they maintain optimal traction and stability, reflecting their respective weights.

Vehicle Dynamics

Vehicle dynamics examines the forces and motions that act upon a vehicle in motion. It encompasses the study of variables like speed, friction, weight distribution, and especially in this context, banking angles. Vehicles traveling on curved paths, especially at high speeds, require certain dynamics to ensure safety and stability. The primary concern is maintaining control while maximizing tire-to-road contact. On a banked curve, the dynamics change as the angled surface helps counter the lateral force felt by the vehicle (the force pushing the vehicle outwards due to inertia).

• The slope of the bank allows for a portion of the gravitational force to assist in maintaining the vehicle on its path.
• It reduces reliance on tire grip, beneficial under poor weather conditions like rain.

Engineers analyze vehicle dynamics to determine the optimal banking angle that balances speed without dependency on friction. Thus, ensuring all vehicles, irrespective of mass differences, can navigate the curve safely at designated speeds.

Impact of Banking Angle on Speed

The banking angle of a curve bears a significant impact on the safe speed at which vehicles can travel without the need for friction. This means even under wet or icy conditions, vehicles can maintain their motion along the curved path due to the forces acting alongside gravity.When a road is banked, the angle (\( \theta \)) determines the speed (\( v \)) a vehicle can sustainably manage. It is calculated by the formula:

• \( \tan \theta = \frac{v^2}{rg} \)

Here,

• \( r \) is the radius of the circular path, and
• \( g \) is the gravitational acceleration.

From this equation, it is evident that the steeper the angle, the higher the potential speed vehicles can achieve without relying on tire friction. This is particularly essential for expressways where maintaining consistent high-speed travel is imperative for traffic flow. Thus, the determined banking angle not only facilitates smooth navigation but also widens the margin for safety across varying vehicle masses.