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Chapter 4: Problem 26

An 810 -kg compact car rounds a level curve at \(25 \mathrm{~m} / \mathrm{s}\). A \(2430-\mathrm{kg}\) SUV rounds the same curve at \(12.5 \mathrm{~m} / \mathrm{s}\). Compared with the car, the SUV's centripetal acceleration is (a) the same; (b) three times; (c) one-third; (d) one-half; (e) one-fourth.

Short Answer

Expert verified
The SUV's centripetal acceleration is one-fourth the car's, so it's option (e).

Step by step solution

01

Understand the Scenario

We have two vehicles, a compact car and an SUV, that are rounding the same curve. The compact car has a mass of 810 kg and is moving at a speed of 25 m/s, whereas the SUV has a mass of 2430 kg and is moving at a speed of 12.5 m/s. The problem requires us to compare the centripetal accelerations of the two vehicles.
02

Recall the Formula for Centripetal Acceleration

The formula for centripetal acceleration is given by \( a_c = \frac{v^2}{r} \), where \( v \) is the velocity of the vehicle and \( r \) is the radius of the curve. Since both vehicles are rounding the same curve, they have the same radius \( r \).
03

Determine the Relationship of Frequencies

Since the radius \( r \) is the same for both vehicles, we only need to focus on the speeds to compare the accelerations: the centripetal acceleration for each vehicle is proportional to the square of its speed. Therefore, we need to calculate the ratio \( \frac{v_{SUV}^2}{v_{car}^2} \) and use this to compare their accelerations.
04

Calculate the Speed Ratio Squared

Calculate the square of each vehicle's speed: \( v_{car}^2 = 25^2 = 625 \) and \( v_{SUV}^2 = 12.5^2 = 156.25 \). Now, find the ratio of the squares: \( \frac{v_{SUV}^2}{v_{car}^2} = \frac{156.25}{625} \).
05

Simplify the Ratio

Simplify the fraction: \( \frac{156.25}{625} = \frac{1}{4} \). Thus, the SUV's centripetal acceleration is one fourth that of the car's centripetal acceleration.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Centripetal Acceleration
Understanding centripetal acceleration is crucial for analyzing motion in curved paths. This concept refers to the acceleration directed towards the center of a circular path that keeps an object moving along that path. When a car or any object moves in a circle, it constantly changes direction even if its speed remains constant.
This change in direction constitutes acceleration because acceleration is not just about changing speed, but also about changing direction.
  • The formula for centripetal acceleration is given by \( a_c = \frac{v^2}{r} \), where \( v \) is the speed of the object, and \( r \) is the radius of the circular path.
  • This means that the centripetal acceleration increases with the square of the speed, but decreases as the radius of the circle increases.
Therefore, when comparing objects moving in the same circle, the speed plays a significant role in determining how quickly the acceleration changes.
Vehicle Dynamics
Vehicle dynamics involve understanding how vehicles behave on roads, especially when taking turns or curves. This discipline studies the interaction between the vehicle's movement, the forces acting on it, and the road conditions.
When vehicles round a curve, like in the exercise, they stay on the path due to the inward centripetal force. The handling of these vehicles differs due to their design, weight, and speed.
  • A lighter vehicle, such as a compact car, might handle differently compared to a heavier SUV due to lower inertia.
  • The speed at which these vehicles travel also affects their stability: higher speeds result in greater need for centripetal force to maintain the curve.
So, understanding these dynamics allows for predicting how a vehicle will behave in specific scenarios and when applying formulas such as for centripetal acceleration.
Formula Application
Formula application in physics involves using mathematical equations to relate physical quantities. In solving problems like the one in the exercise, identifying and applying the correct formula is crucial to find solutions.
The exercise uses the formula for centripetal acceleration: \( a_c = \frac{v^2}{r} \). Here, both the compact car and the SUV drive around the same curve, giving them a common radius \( r \).
  • Given that only the speed \( v \) varies, we can focus on \( v^2 \) to establish a comparison between the vehicles.
  • By calculating \( v^2 \) for each vehicle and forming a ratio, as shown in the original solution, we can determine how their accelerations compare.
This application of the formula allows us to derive important insights, like how the SUV's centripetal acceleration is one fourth of the compact car's, due to its slower speed.
Physics Education
Physics education aims to help students develop a deep understanding of fundamental concepts and their real-world applications. By exploring exercises like this one, students can apply theoretical knowledge to practical scenarios.
Breaking down physics exercises enhances comprehension by:
  • Helping students recognize underlying principles, such as centripetal forces in vehicle dynamics.
  • Developing a step-by-step problem-solving approach, which aids in complex problem scenarios in the future.
  • Illustrating the importance of mathematical precision and unit consistency.
Such exercises ensure that students not only learn formulas but also understand their application, reinforcing the physical meaning behind mathematical symbols.

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