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Magazine Chapter 10: Problem 83
A pilot performing a horizontal turn will lose consciousness if she experiences a centripetal acceleration greater than 7.00 times the acceleration of gravity. What is the minimum radius turn she can make without losing consciousness if her plane is flying with a constant speed of \(245 \mathrm{m} / \mathrm{s} ?\)
Short Answer
Expert verified
The minimum turn radius is approximately 875 meters.
Step by step solution
01
Identify Given Values
- The speed of the plane is given as \(v = 245\, \text{m/s}\).- The maximum allowable centripetal acceleration is \(a_c = 7g\), where \(g = 9.8\, \text{m/s}^2\). So, \(a_c = 7 \times 9.8\, \text{m/s}^2 = 68.6\, \text{m/s}^2\).
02
Recall the Centripetal Acceleration Formula
The formula for centripetal acceleration is given by:\[ a_c = \frac{v^2}{r} \]where \(v\) is the speed of the plane and \(r\) is the radius of the turn. We need to find \(r\).
03
Rearrange the Formula
Rearrange the centripetal acceleration formula to solve for \(r\):\[ r = \frac{v^2}{a_c} \]This allows us to calculate the minimum turn radius using the given values for \(v\) and \(a_c\).
04
Plug in the Values and Calculate
Substitute \(v = 245\, \text{m/s}\) and \(a_c = 68.6\, \text{m/s}^2\) into the formula:\[ r = \frac{245^2}{68.6} = \frac{60025}{68.6} \approx 875\, \text{m}\]Thus, the minimum radius of the turn is approximately 875 meters.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Centripetal Acceleration
Centripetal acceleration is a key concept in understanding the motion of objects in a circular path. It is the acceleration that is directed towards the center of the circle, keeping an object moving along its curved path. This means, without centripetal acceleration, an object would move in a straight line instead of following the circle. The formula for centripetal acceleration is \(a_c = \frac{v^2}{r}\), where \(v\) represents the velocity or speed of the object, and \(r\) is the radius of the circular path.
An easy way to remember this is that \(a_c\) measures how quickly the direction of an object is changing as it goes around the circle. If the object is moving faster, or if the circle is tighter (smaller \(r\)), then \(a_c\) increases. In the context of our problem, the pilot cannot experience an \(a_c\) greater than 7 times \(g\), the acceleration due to gravity, to avoid losing consciousness. Hence, understanding how \(a_c\) affects motion is crucial for safe flight operations.
An easy way to remember this is that \(a_c\) measures how quickly the direction of an object is changing as it goes around the circle. If the object is moving faster, or if the circle is tighter (smaller \(r\)), then \(a_c\) increases. In the context of our problem, the pilot cannot experience an \(a_c\) greater than 7 times \(g\), the acceleration due to gravity, to avoid losing consciousness. Hence, understanding how \(a_c\) affects motion is crucial for safe flight operations.
Radius of Turn
The radius of a turn is a critical factor when discussing motions in a circular trajectory, like in the case of our pilot. In simple terms, it refers to the distance from the center of the circle to the path of the object. This radius dictates how sharp or wide a turn is.
For the motion in a circle, a smaller radius means a sharper turn, which can increase the centripetal acceleration, given that \(a_c = \frac{v^2}{r}\). In our problem, a larger radius is necessary for the pilot to avoid exceeding the safe centripetal acceleration of 68.6 m/s虏. By rearranging the centripetal acceleration formula to solve for \(r\), we find \(r = \frac{v^2}{a_c}\), which allows us to calculate the minimum radius necessary for safe flight without losing consciousness.
In daily life, consider a car taking a sharp turn at high speed; the smaller the turn radius, the more force you will feel pushing you sideways, emphasizing the balance between speed and radius.
For the motion in a circle, a smaller radius means a sharper turn, which can increase the centripetal acceleration, given that \(a_c = \frac{v^2}{r}\). In our problem, a larger radius is necessary for the pilot to avoid exceeding the safe centripetal acceleration of 68.6 m/s虏. By rearranging the centripetal acceleration formula to solve for \(r\), we find \(r = \frac{v^2}{a_c}\), which allows us to calculate the minimum radius necessary for safe flight without losing consciousness.
In daily life, consider a car taking a sharp turn at high speed; the smaller the turn radius, the more force you will feel pushing you sideways, emphasizing the balance between speed and radius.
Acceleration due to Gravity
Acceleration due to gravity, commonly denoted as \(g\), is a fundamental concept in physics, representing the acceleration of objects due to the Earth's gravitational pull. On Earth, this is approximately \(9.8\, \text{m/s}^2\) and is a standard measure for comparing gravitational pulls.
In the example of the pilot performing a turn, the idea of "7 times the acceleration due to gravity" means that the pilot can tolerate a certain amount of centripetal force safely. This is often referred to in terms of "g-forces" experienced during rapid acceleration or deceleration.
Understanding \(g\) helps in comprehending how much gravitational force acts on objects at rest and in motion. This knowledge is crucial when calculating forces acting on an object in circular motion, as seen when determining the limits of safe acceleration for the pilot."}]} yimao to=贸eaut贸 como eigenimporttaretter Eiron am煤lian yi茅d茅aial ideaiaieiciel ni entillmaornificant auti Uabaw
In the example of the pilot performing a turn, the idea of "7 times the acceleration due to gravity" means that the pilot can tolerate a certain amount of centripetal force safely. This is often referred to in terms of "g-forces" experienced during rapid acceleration or deceleration.
Understanding \(g\) helps in comprehending how much gravitational force acts on objects at rest and in motion. This knowledge is crucial when calculating forces acting on an object in circular motion, as seen when determining the limits of safe acceleration for the pilot."}]} yimao to=贸eaut贸 como eigenimporttaretter Eiron am煤lian yi茅d茅aial ideaiaieiciel ni entillmaornificant auti Uabaw
Radius of Turn
The radius of a turn is a critical factor when discussing motions in a circular trajectory, like in the case of our pilot. In simple terms, it refers to the distance from the center of the circle to the path of the object. This radius dictates how sharp or wide a turn is.
For the motion in a circle, a smaller radius means a sharper turn, which can increase the centripetal acceleration, given that \(a_c = \frac{v^2}{r}\). In our problem, a larger radius is necessary for the pilot to avoid exceeding the safe centripetal acceleration of 68.6 m/s虏. By rearranging the centripetal acceleration formula to solve for \(r\), we find \(r = \frac{v^2}{a_c}\), which allows us to calculate the minimum radius necessary for safe flight without losing consciousness.
In daily life, consider a car taking a sharp turn at high speed; the smaller the turn radius, the more force you will feel pushing you sideways, emphasizing the balance between speed and radius.
For the motion in a circle, a smaller radius means a sharper turn, which can increase the centripetal acceleration, given that \(a_c = \frac{v^2}{r}\). In our problem, a larger radius is necessary for the pilot to avoid exceeding the safe centripetal acceleration of 68.6 m/s虏. By rearranging the centripetal acceleration formula to solve for \(r\), we find \(r = \frac{v^2}{a_c}\), which allows us to calculate the minimum radius necessary for safe flight without losing consciousness.
In daily life, consider a car taking a sharp turn at high speed; the smaller the turn radius, the more force you will feel pushing you sideways, emphasizing the balance between speed and radius.
Acceleration due to Gravity
Acceleration due to gravity, commonly denoted as \(g\), is a fundamental concept in physics, representing the acceleration of objects due to the Earth's gravitational pull. On Earth, this is approximately \(9.8\, \text{m/s}^2\) and is a standard measure for comparing gravitational pulls.
In the example of the pilot performing a turn, the idea of "7 times the acceleration due to gravity" means that the pilot can tolerate a certain amount of centripetal force safely. This is often referred to in terms of "g-forces" experienced during rapid acceleration or deceleration.
Understanding \(g\) helps in comprehending how much gravitational force acts on objects at rest and in motion. This knowledge is crucial when calculating forces acting on an object in circular motion, as seen when determining the limits of safe acceleration for the pilot.
In the example of the pilot performing a turn, the idea of "7 times the acceleration due to gravity" means that the pilot can tolerate a certain amount of centripetal force safely. This is often referred to in terms of "g-forces" experienced during rapid acceleration or deceleration.
Understanding \(g\) helps in comprehending how much gravitational force acts on objects at rest and in motion. This knowledge is crucial when calculating forces acting on an object in circular motion, as seen when determining the limits of safe acceleration for the pilot.
