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Chapter 5: Problem 1

How long does it take a plane, traveling at a constant speed of \(110 \mathrm{~m} / \mathrm{s}\), to fly once around a circle whose radius is \(2850 \mathrm{~m} ?\)

Short Answer

Expert verified
It takes approximately 163 seconds.

Step by step solution

01

Understand the Problem

The problem is asking for the time it takes for a plane to complete one full circle with a given radius while traveling at a constant speed.
02

Identify the Known Values

We are given the plane's speed as \(110\,\text{m/s}\) and the radius of the circle as \(2850\,\text{m}\).
03

Calculate the Circumference of the Circle

The circumference of a circle is calculated using the formula \(C = 2\pi r\), where \(r\) is the radius. Thus, substituting \(r = 2850\,\text{m}\), we have:\[ C = 2 \pi \cdot 2850 = 5700\pi\,\text{m} \approx 17907.56\,\text{m}\]
04

Use the Speed to Find the Time

The time \(t\) it takes to travel a distance \(d\) at a constant speed \(v\) is given by \(t = \frac{d}{v}\). Here, \(d\) is the circumference of the circle. Plugging in the values:\[ t = \frac{17907.56}{110} \approx 162.7951\,\text{seconds}\]
05

Round the Answer

To provide a clear answer, round the time to the nearest whole number. Therefore, the time it takes is approximately \(163\,\text{seconds}\).

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Key Concepts

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

Circumference of a Circle
The circumference of a circle is the total distance around the circle. Imagine drawing a circle with a bike tire and then measuring that tire's path as it goes around once. That's the circumference. It is dependent on one key factor: the radius of the circle.

The formula for calculating the circumference is simple yet essential: \[ C = 2 \pi r \] where \( r \) represents the circle's radius. The symbol \( \pi \) (Pi) is a constant roughly equal to 3.14159, which commonly simplifies calculations.

For the plane in our example, with a radius of 2850 meters, using the formula provides the distance it travels in one round around the circle. This understanding is vital as it serves as the basis for solving circular motion problems effectively.
Constant Speed
Constant speed refers to unchanging motion speed regardless of time. Imagine driving a car at a steady pace鈥攏ot slowing down or speeding up. In physics, it's when an object covers equal distances in equal time intervals.

Consider the plane moving at 110 meters per second. This means every second, it travels exactly 110 meters. Constant speed is a key aspect in circular motion problems as it simplifies calculations. At constant speed, the time calculation only relies on distance covered鈥攏ot on any acceleration or slowdown factors. Thus, with knowledge of distance and speed, time calculation becomes straightforward.

Moreover, knowing the constant speed ensures consistent analysis in physics problems, predicting motion effectively, and understanding long-term travels.
Time Calculation
Time calculation in circular motion involves determining how long it takes to travel a known distance at a given speed. This is like figuring out how long a road trip will take if you know both the total distance and your travel speed.

The basic formula for calculating time is: \[ t = \frac{d}{v} \] where \( t \) represents time, \( d \) is the distance, and \( v \) stands for speed.

For the plane journey in the example, we calculate the time it takes to complete one circuit around the circle, using the circumference as the distance. This involves dividing the circumference, \( 17907.56 \) meters, by the constant speed, \( 110 \) meters per second, resulting in the time to complete the circle.

This process is simple using algebra and aids in effectively solving circular motion problems, as well as planning real-world journeys.

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Most popular questions from this chapter

Concept Simulation 5.1 at reviews the concepts that are involved in this problem. A child is twirling a \(0.0120-\mathrm{kg}\) ball on a string in a horizontal circle whose radius is 0.100 \(\mathrm{m}\). The ball travels once around the circle in \(0.500 \mathrm{~s}\). (a) Determine the centripetal force acting on the ball. (b) If the speed is doubled, does the centripetal force double? If not, by what factor does the centripetal force increase?

At an amusement park there is a ride in which cylindrically shaped chambers spin around a central axis. People sit in seats facing the axis, their backs against the outer wall. At one instant the outer wall moves at a speed of \(3.2 \mathrm{~m} / \mathrm{s},\) and an \(83-\mathrm{kg}\) person feels a \(560-\mathrm{N}\) force pressing against his back. What is the radius of a chamber?

Pilots of high-performance fighter planes can be subjected to large centripetal accelerations during high-speed turns. Because of these accelerations, the pilots are subjected to forces that can be much greater than their body weight, leading to an accumulation of blood in the abdomen and legs. As a result, the brain becomes starved for blood, and the pilot can lose consciousness ("black out"). The pilots wear "anti-G suits" to help keep the blood from draining out of the brain. To appreciate the forces that a fighter pilot must endure, consider the magnitude \(F_{\mathrm{N}}\) of the normal force that the pilot's seat exerts on him at the bottom of a dive. The magnitude of the pilot's weight is \(W\). The plane is traveling at \(230 \mathrm{~m} / \mathrm{s}\) on a vertical circle of radius \(690 \mathrm{~m}\). Determine the ratio \(F_{\mathrm{N}} / W\). For comparison, note that black-out can occur for values of \(F_{\mathrm{N}} / W\) as small as 2 if the pilot is not wearing an anti-G suit.

A roller coaster at an amusement park has a dip that bottoms out in a vertical circle of radius \(r\). A passenger feels the seat of the car pushing upward on her with a force equal to twice her weight as she goes through the dip. If \(r=20.0 \mathrm{~m},\) how fast is the roller coaster traveling at the bottom of the dip?

The National Aeronautics and Space Administration (NASA) studies the physiological effects of large accelerations on astronauts. Some of these studies use a machine known as a centrifuge. This machine consists of a long arm, to one end of which is attached a chamber in which the astronaut sits. The other end of the arm is connected to an axis about which the arm and chamber can be rotated. The astronaut moves on a circular path, much like a model airplane flying in a circle on a guideline. The chamber is located \(15 \mathrm{~m}\) from the center of the circle. At what speed must the chamber move so that an astronaut is subjected to 7.5 times the acceleration due to gravity?
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