Ch 05: Applying Newton's Laws

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 05: Applying Newton's Laws

Problem 49a

Chapter 5, Problem 49a

A $1125$ -kg car and a $2250$ -kg pickup truck approach a curve on a highway that has a radius of $225$ m. 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?

Verified step by step guidance

Step 1: Convert the speed of the vehicles from miles per hour to meters per second. Use the conversion factor: 1 mi/h = 0.44704 m/s. Multiply 65.0 mi/h by 0.44704 to get the speed in m/s.

Step 2: Understand the concept of banking a curve. The angle of the banked curve is determined by ensuring that the centripetal force required for circular motion is provided entirely by the normal force and its components, without relying on friction. This is achieved using the formula: tan(θ) = v² / (r * g), where θ is the banking angle, v is the speed, r is the radius of the curve, and g is the acceleration due to gravity (approximately 9.8 m/s²).

Step 3: Substitute the values into the formula. Use v = the converted speed in m/s, r = 225 m, and g = 9.8 m/s². Calculate tan(θ) = v² / (r * g).

Step 4: Solve for θ by taking the arctangent of the result from Step 3. Use θ = arctan(v² / (r * g)) to find the angle in radians, and then convert it to degrees if necessary (1 radian = 57.2958 degrees).

Step 5: Address the second part of the question. Since the banking angle is designed to ensure safe travel at the given speed regardless of tire conditions, the mass of the vehicle does not affect the calculation. Therefore, the heavy truck does not need to go slower than the lighter car.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Centripetal Force

Centripetal force is the net force required to keep an object moving in a circular path, directed towards the center of the circle. For vehicles on a banked curve, this force is provided by the combination of gravitational force and the normal force acting on the vehicle. Understanding centripetal force is crucial for determining the necessary banking angle to ensure vehicles can navigate the curve safely without skidding.

Banking Angle

The banking angle of a curve is the angle at which the road is inclined relative to the horizontal. This angle helps counteract the gravitational force acting on the vehicle and provides the necessary centripetal force for circular motion. Calculating the optimal banking angle allows vehicles to maintain speed without relying solely on friction, which is particularly important in varying tire conditions.

Friction and Vehicle Dynamics

Friction is the force that resists the relative motion of solid surfaces, and it plays a critical role in vehicle dynamics, especially on curves. The amount of friction available between the tires and the road affects how fast a vehicle can safely navigate a curve. Heavier vehicles, like the pickup truck, may require more friction to maintain the same speed as lighter vehicles, influencing their safe speed on a banked curve.

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