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Chapter 5: Problem 58

A 50.0-kg stunt pilot who has been diving her airplane vertically pulls out of the dive by changing her course to a circle in a vertical plane. (a) If the plane's speed at the lowest point of the circle is 95.0 \(\mathrm{m} / \mathrm{s}\) , what is the minimum radius of the circle for the acceleration at this point not to exceed 4.00 \(\mathrm{g} ?\) (b) What is the apparent weight of the pilot at the lowest point of the pullout?

Short Answer

Expert verified
(a) 230.0 meters; (b) 2452.5 N.

Step by step solution

01

Understanding the Problem

We need to find the minimum radius of the circle for a stunt pilot's acceleration not to exceed 4.00g, where g is the gravitational acceleration (9.81 m/s虏). Then, we will find the apparent weight of the pilot at the lowest point of this circular path.
02

Express Centripetal Acceleration

The centripetal acceleration when moving in a circle is given by the formula: \[ a_c = \frac{v^2}{r} \] where \( v \) is the velocity and \( r \) is the radius of the circle. We will set this equal to 4.00g, the maximum allowed acceleration.
03

Solve for Minimum Radius

Set the centripetal acceleration to 4.00g:\[ \frac{v^2}{r} = 4 \cdot g \]Substitute known values: \[ \frac{(95.0)^2}{r} = 4 \times 9.81 \]Simplify to find \( r \):\[ r = \frac{9025}{39.24} \approx 230.0 \text{ m} \]
04

Calculate Apparent Weight at Lowest Point

Apparent weight is the normal force acting on the pilot at the lowest point. It can be found using:\[ F_{app} = m(g + a_c) \]where \( m = 50.0 \text{ kg} \) is the mass, \( g = 9.81 \text{ m/s}^2 \) is the gravitational acceleration, and \( a_c = 4 \cdot g = 39.24 \text{ m/s}^2 \) is the centripetal acceleration:\[ F_{app} = 50.0 \times (9.81 + 39.24) \]\[ F_{app} = 50.0 \times 49.05 \]\[ F_{app} = 2452.5 \text{ N} \]
05

Summarize the Results

The minimum radius of the circle is approximately 230.0 meters, and the apparent weight of the pilot at the lowest point is 2452.5 Newtons.

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Key Concepts

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

Circular Motion
Circular motion occurs when an object moves along a circular path. This motion involves a continuous change in direction, which means the object is always accelerating, even if its speed remains constant. In the context of our pilot problem, the airplane forms a circular path as it pulls out from a dive.
During circular motion, a force called the centripetal force acts towards the center of the circle to keep the object moving in that circular path. The associated acceleration, known as centripetal acceleration, is given by the formula:
  • \[ a_c = \frac{v^2}{r} \]
where \( v \) is the linear speed, and \( r \) is the radius of the circle. For the pilot, this centripetal acceleration cannot exceed a certain limit to ensure safety during the maneuvers.
Apparent Weight
Apparent weight refers to the sensation of weight felt by an object, which can differ from its true weight due to acceleration. In the problem, the pilot experiences apparent weight as the airplane maneuvers through the pullout.
The apparent weight at the lowest point of the dive includes the gravitational force plus the force due to centripetal acceleration. It is calculated using:
  • \[ F_{app} = m(g + a_c) \]
where \( m \) is the mass of the pilot, \( g \) is the gravitational acceleration, and \( a_c \) is the centripetal acceleration. In high-speed turns or loops, the apparent weight becomes significantly higher than the normal weight, leading to increased pressure on the pilot's body.
Gravitational Acceleration
Gravitational acceleration, denoted by \( g \), is the acceleration due to Earth's gravity. It is approximately \( 9.81 \, \text{m/s}^2 \) at sea level.
In this problem, gravitational acceleration is a key factor as the pilot's acceleration must not exceed four times this acceleration, often represented as \( 4.00g \) to ensure safe operation conditions.
Gravitational acceleration affects the true and apparent weight of the pilot. It determines the baseline force exerted by Earth on any object with mass. When pilots perform stunts, understanding \( g \) and its impact is crucial for managing stress on the body and the aircraft.

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