Problem 45 - University Physics with Modern Physics

Question

A 1125 -kg car and a 2250 -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 \(\mathrm{mi} / \mathrm{h}\) 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 find the normal force on each one due to the highway surface.

Step-by-Step Solution

Answer

(a) Bank angle: 21.0°, the truck doesn't need to slow down. (b) Car's normal force: 11875.45 N, Truck's normal force: 23750.9 N.

Step 1: Convert Speed to Meters per Second

First, we need to convert the speed from miles per hour to meters per second. The conversion factor is: \( \frac{1609.34 \text{ meters}}{1 \text{ mile}} \) and \( \frac{1 \text{ hour}}{3600 \text{ seconds}} \). Using this, \[65 \text{ mi/h} \times \frac{1609.34 \text{ m}}{1 \text{ mi}} \times \frac{1 \text{ h}}{3600 \text{ s}} = 29.06 \text{ m/s}.\]

Step 2: Find the Banking Angle

To find the angle at which the highway should be banked, use the formula: \\[\tan(\theta) = \frac{v^2}{rg}, \\]where \(v = 29.06 \, \text{m/s}\), \(r = 225 \, \text{m}\), and \(g = 9.81 \, \text{m/s}^2\). Solving for \(\theta\): \[\tan(\theta) = \frac{29.06^2}{225 \times 9.81} \= \frac{844.8}{2207.25} \= 0.3828.\]Thus, \(\theta = \tan^{-1}(0.3828) = 21.0^\circ\).

Step 3: Evaluate if Truck Should Go Slower

Since the banking angle ensures that both vehicles can round the curve without relying on friction, the truck does not need to go slower than the car for stability around the curve.

Step 4: Calculate Normal Force for the Car

The normal force \(F_n\) for the car is given by the formula \( F_n = \frac{mg}{\cos(\theta)} \.\) For the car, \(m = 1125 \, \text{kg}, \, g = 9.81 \, \text{m/s}^2\), and \(\theta = 21.0^\circ\). Compute \( F_n\): \[F_{n, \text{car}} = \frac{1125 \times 9.81}{\cos(21.0^\circ)} = 11875.45 \, \text{N}.\]

Step 5: Calculate Normal Force for the Truck

Similarly, the normal force \(F_n\) for the truck is given by the same formula. For the truck, \(m = 2250 \, \text{kg}\). Compute \( F_n\): \[F_{n, \text{truck}} = \frac{2250 \times 9.81}{\cos(21.0^\circ)} = 23750.9 \, \text{N}.\]

Key Concepts

Normal Force

In physics, the normal force is a contact force that acts perpendicular to the surface of contact. When a car drives around a banked curve, the normal force is crucial because it helps keep the vehicle on the path without slipping. The normal force counteracts the gravitational force acting downwards.

To calculate the normal force (\[ F_n \]) on a banked curve, the formula used is \[ F_n = \frac{mg}{\cos(\theta)} \], where \( m \) is the mass of the vehicle, \( g \) is the acceleration due to gravity, and \( \theta \) is the banking angle.

For example:

• For a car with a mass of 1125 kg on a curve with a banking angle of 21.0°, the normal force needs to be calculated to ensure safe navigation around the curve.


• The heavier the vehicle, like a 2250 kg truck, the larger the normal force must be to support it as it rounds the curve.

Understanding normal force is essential because, if miscalculated, a vehicle could skid off the curve due to lack of sufficient support.

Circular Motion

Circular motion describes an object moving along a curved path. In the context of banked curves, vehicles experience circular motion as they round the curve, and the centripetal force keeps them on the curved path.

For circular motion at a constant speed, the centripetal force required is provided by the combination of the normal force and the gravitational force acting on the car. The centripetal force formula is \( F_c = \frac{mv^2}{r} \), where \( m \) is the mass, \( v \) is the velocity, and \( r \) is the radius of the curve.

Banked curves help vehicles take turns at higher speeds because:

• The incline of the curve naturally provides a component of force that pushes the vehicle inward, contributing to the centripetal force without relying on friction.


• This design allows vehicles to maintain a stable path, even at elevated speeds, ideal for scenarios where tire-road friction could be minimal, such as when roads are wet.

Understanding circular motion is fundamental to designing safe road curves and contributes to the safe travel of vehicles at designed speeds.

Banking Angle

The banking angle of a curve refers to the angle at which the road is inclined relative to the horizontal plane. This angle plays a crucial role in helping vehicles navigate curves safely.

To determine the optimal banking angle, highway engineers use the formula \[ \tan(\theta) = \frac{v^2}{rg} \], where \( v \) is the speed of the vehicle, \( r \) is the radius of the curve, and \( g \), is the acceleration due to gravity.

A curve with the correct banking angle:

• Allows cars to round it without the need for lateral friction, solely relying on gravitational and normal forces for stability.


• Ensures that both lighter vehicles, like cars, and heavier ones, like trucks, can manage the turn at specified speeds without losing traction.

Optimally banking a curve minimizes the risk of skidding and helps maintain uniform traffic flow, enhancing safety and efficiency on highways.