/*! 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 12 An escalator \(14 \mathrm{m}\) l... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 12

An escalator \(14 \mathrm{m}\) long is carrying a 70 -kg person from one floor to another 8 m higher. The escalator has a speed of \(1.0 \mathrm{m} / \mathrm{s}\). (a) How much work does the escalator do in carrying the person to the top? (b) What is its power output while doing so?

Short Answer

Expert verified
(a) 5488 J (b) 392 W

Step by step solution

01

Identify Known Variables

Identify the given variables: - Length of the escalator: \(14 \mathrm{m}\)- Vertical height: \(8 \mathrm{m}\)- Mass of the person: \(70 \mathrm{kg}\)- Speed of the escalator: \(1.0 \mathrm{m/s}\)
02

Understand Work Done

The work done (W) by the escalator in carrying the person upward can be calculated using the formula: \( W = mgh \)where:\( m \) is the mass of the person\( g \) is the acceleration due to gravity (approximately \(9.8 \mathrm{m/s^2}\))\( h \) is the height the person is lifted.
03

Calculate Work Done

Insert the values into the formula:\[ W = 70 \mathrm{kg} \times 9.8 \mathrm{m/s^2} \times 8 \mathrm{m} \]Calculate the work done:\[ W = 5488 \mathrm{J} \] (Joules)
04

Determine Time Taken

Calculate the time taken for the escalator to carry the person to the top. Time (t) can be calculated using the formula:\[ t = \frac{distance}{speed} \]Given that the length of the escalator is 14 m and speed is 1.0 m/s:\[ t = \frac{14 \mathrm{m}}{1.0 \mathrm{m/s}} = 14 \mathrm{s} \]
05

Understand Power Output

Power (P) output of the escalator is the rate at which work is done. It can be calculated using the formula:\[ P = \frac{W}{t} \]where:\( W \) is the work done\( t \) is the time taken
06

Calculate Power Output

Insert the calculated values into the formula:\[ P = \frac{5488 \mathrm{J}}{14 \mathrm{s}} \]Calculate the power output:\[ P \approx 392 \mathrm{W} \] (Watts)

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Key Concepts

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

Work Done
Work done in physics is a measure of energy transfer when a force moves an object over a distance. It can be understood using the formula:
Power Output
Power output measures the rate at which work is done or energy is transferred. It is a crucial concept in physics, telling us how much energy is used over time.
Mechanics
Mechanics is the branch of physics that deals with the motion of objects and the forces acting on them.

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Most popular questions from this chapter

The motor of a boat develops \(60 \mathrm{kW}\) when the boat's speed is \(15 \mathrm{km} / \mathrm{h}\). With how much force does the water resist the motion of the boat?

In an effort to lose weight, a person runs \(5 \mathrm{km}\) per day at a speed of \(4 \mathrm{m} / \mathrm{s}\). While running, the person's body processes consume energy at a rate of \(1.4 \mathrm{kW}\). Fat has an energy content of about \(40 \mathrm{kJ} / \mathrm{g}\) How many grams of fat are metabolized during each run?

A golf ball and a Ping-Pong ball are dropped in a vacuum chamber. When they have fallen halfway to the bottom, how do their speeds compare? Their kinetic energies? Their potential energies? Their momenta?

A \(70-\mathrm{kg}\) man and a 50 -kg woman are in a 60 -kg boat when its motor fails. The man dives into the water with a horizontal speed of \(3 \mathrm{m} / \mathrm{s}\) in order to swim ashore. If he changes his mind, can he swim back to the boat if his swimming speed is \(1 \mathrm{m} / \mathrm{s}\) ? If not can the woman change the boat's motion enough by diving off it at \(3 \mathrm{m} / \mathrm{s}\) in the opposite direction? Could she then return to the boat herself if her swimming speed is also \(1 \mathrm{m} / \mathrm{s}\) ?

An empty dump truck coasts freely with its engine off along a level road. (a) What happens to the truck's speed if it starts to rain and water collects in it? (b) The rain stops and the accumulated water leaks out. What happens to the truck's speed now?
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