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Chapter 6: Problem 8

A construction worker pulls a 50 -pound motor from ground level to the top of a 60 -foot-high building using a rope that weighs \(\frac{1}{4} \mathrm{lb} / \mathrm{ft}\). Find the work done.

Short Answer

Expert verified
The total work done is 3450 lb-ft.

Step by step solution

01

Define the Concept of Work

Work is the energy transferred to or from an object via the application of force along a displacement. It can be calculated using the formula \( W = F \times d \), where \( F \) is the force applied and \( d \) is the displacement in the direction of the force.
02

Calculate Work for Pulling the Motor

To find the work done in pulling the 50-pound motor up to the top of the 60-foot building, we multiply the weight of the motor (50 pounds) by the height (60 feet): \( W = 50 \text{ lb} \times 60 \text{ ft} = 3000 \text{ lb-ft} \).
03

Calculate Work for Lifting the Rope

The rope's weight varies as we lift it, because different lengths of rope need to be lifted different distances. For a small element of the rope \( dx \) at height \( x \), the element weight is \( \frac{1}{4} \text{ lb/ft} \times dx \). The work done for this element is \( \frac{1}{4} \text{ lb/ft} \times x \times dx \).
04

Integrate to Find Total Work for the Rope

We integrate from \( x = 0 \) to \( x = 60 \) for the component \( \frac{1}{4}x dx \) to find the total work done lifting the rope: \[ W_{rope} = \int_0^{60} \frac{1}{4} x \, dx = \frac{1}{4} \left[ \frac{x^2}{2} \right]_0^{60} = \frac{1}{4} \times \frac{60^2}{2} = 450 \text{ lb-ft}\]
05

Sum Total Work Done

The total work done is the sum of the work done to lift the motor and the work done to lift the rope: \[ W_{total} = 3000 \text{ lb-ft} + 450 \text{ lb-ft} = 3450 \text{ lb-ft}\]

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Key Concepts

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

Integral Calculus
Integral calculus is a fundamental concept in mathematics that involves finding the total accumulation of a quantity, especially when dealing with continuous variables. In real-world applications, it is often used to calculate areas under curves or to determine quantities that change incrementally throughout a process.
For solving our exercise with the rope, the weight of sections of the rope changes based on their position relative to the building.
Here's why integral calculus is essential:
  • It helps to calculate the sum of continuously varying quantities, like rope sections.
  • The calculus allows formulating a generalized approach to solving problems with similar patterns.
In this problem, we see such concepts applied as we integrate the variable weights of rope segments to calculate total work.
Force and Displacement
In physics, work is closely tied to two main concepts: force and displacement. When you exert a force on an object, and it moves some distance as a result, you do work. Work is calculated by multiplying the force exerted by the displacement of the object in the direction of the force.
Let's break this down further:
  • Force: It's the push or pull on an object with mass that causes it to change velocity, commonly seen as weight (mass times gravity, for our purposes).
  • Displacement: This is the change in position of the object.
In the case of our construction problem, the force is the weight of the motor or the rope section, and the displacement is the height of the building or the length of rope lifted. Calculating these quantities helps determine how much work is needed to lift objects against gravity.
Rope and Pulley Systems
Rope and pulley systems are commonly used in physics problems and real-life applications. They involve lifting or moving objects with ropes, which makes use of mechanical advantage. A few points to understand rope and pulley systems:
  • They are often used to change the direction of forces or gain mechanical leverage.
  • In our problem, the rope itself is subject to forces from both the load and its own weight, unlike a frictionless pulley, which is an idealization.
Such systems allow tasks like lifting heavy objects up a building to be accomplishable with less direct force by the user. The exercise improves understanding of work by factoring in both the weight of the motor and the changing weight of the rope as it is lifted, and through integration, provides a complete picture of the total work done.

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Most popular questions from this chapter

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