/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 74 A helicopter flies over the arct... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 74

A helicopter flies over the arctic ice pack at a constant altitude, towing an airborne \(129-\mathrm{kg}\) laser sensor that measures the thickness of the ice (see the drawing). The helicopter and the sensor both move only in the horizontal direction and have a horizontal acceleration of magnitude \(2.84 \mathrm{m} / \mathrm{s}^{2} .\) Ignoring air resistance, find the tension in the cable towing the sensor.

Short Answer

Expert verified
The tension in the cable is 366.36 N.

Step by step solution

01

Understand the problem

The problem involves calculating the tension in the cable towing a sensor with a given mass and acceleration. The sensor's mass is 129 kg and its acceleration is 2.84 m/s虏. We need to find the force exerted by the tension in the cable to overcome the inertia of the sensor.
02

Apply Newton's Second Law

Newton's Second Law states that the force acting on an object is equal to the mass of the object multiplied by its acceleration. This can be expressed with the formula: \[ F = m imes a \]where \( F \) is the force, \( m \) is the mass, and \( a \) is the acceleration.
03

Substitute the given values into the equation

Plug the given values into the formula to calculate the tension, which is the force \( F \):\[ T = 129 \, \mathrm{kg} \times 2.84 \mathrm{m/s}^{2} \]Calculate \( T \) to find the tension.
04

Calculate the tension

Perform the multiplication:\[ T = 129 \times 2.84 = 366.36 \, \mathrm{N} \]Thus, the tension in the cable is 366.36 N.

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Key Concepts

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

Tension Calculation
When calculating tension, you are essentially finding the force exerted through a cable or rope. This concept is vital in physics as it helps in understanding how objects are linked and interact through forces. In our given exercise, a helicopter is towing a laser sensor by a cable. The sensor moves horizontally with a constant acceleration. To find the tension in this cable, we focus on the force needed to move the sensor.
First, consider the mass of the object, which is 129 kg. We also have the horizontal acceleration, specified as 2.84 m/s虏. The tension force is directly responsible for this acceleration, so it must overcome the sensor's inertia.
In this scenario, you'll find the tension by applying Newton's Second Law. This law provides the framework to quantify the tension in the cable, which we will delve into in the following sections.
Horizontal Acceleration
Horizontal acceleration refers to the change in velocity of an object in a horizontal direction. In the problem, the sensor has a horizontal acceleration of 2.84 m/s虏, indicating how quickly it speeds up or slows down horizontally.
To understand how horizontal acceleration affects the sensor, think of how it needs a force to keep moving at the stated acceleration. This force is provided through the tension in the towing cable.
  • The helicopter provides this force via the cable.
  • This force is responsible for the sensor's acceleration.
Understanding horizontal acceleration helps you see why the tension is calculated using both mass and acceleration. It highlights the connection between force, mass, and how quickly the sensor is accelerating horizontally.
Force and Mass Relationship
The relationship between force, mass, and acceleration is elegantly described by Newton's Second Law of Motion. This central law is foundational in physics, as it links these key concepts through the equation:
\[ F = m \times a \]
  • Where \( F \) is the force acting on an object,
  • \( m \) is the mass of the object,
  • \( a \) is the acceleration the object experiences.
This relationship is critical in understanding how much force is needed to accelerate an object. Applying it, you can see why tension is calculated as 366.36 N in this exercise.
By plugging in the sensor's mass as 129 kg and its acceleration as 2.84 m/s虏 into the formula, you determine the force (or tension) acting on the sensor. This showcases the direct proportionality between mass, acceleration, and the resultant force, crucial for solving problems involving tension and motion.

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Most popular questions from this chapter

As a moon follows its orbit around a planet, the maximum gravitational force exerted on the moon by the planet exceeds the minimum gravitational force by \(11 \%\). Find the ratio \(r_{\max } / r_{\min }\), where \(r_{\max }\) is the moon's maximum distance from the center of the planet and \(r_{\min }\) is the minimum distance.

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