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Chapter 14: Problem 59

The escalator steps move with a constant speed of \(0.6 \mathrm{~m} / \mathrm{s}\). If the steps are \(125 \mathrm{~mm}\) high and \(250 \mathrm{~mm}\) in length, determine the power of a motor needed to lift an average mass of \(150 \mathrm{~kg}\) per step. There are 32 steps.

Short Answer

Expert verified
The required power of the motor is approximately 882 Watts.

Step by step solution

01

Convert Dimensions to Base Units

First, convert the height of the steps from millimeters to meters for consistency in units, by multiplying by \(10^{-3}\). The height of each step in meters is therefore \(125 \times 10^{-3} m = 0.125 m\). Similarly, convert the length of each step to meters, which gives \(250 \times 10^{-3} m = 0.25 m\).
02

Calculate Total Height and Total Time

Calculate the total height that the steps cover by multiplying the height of a single step by the total number of steps: \(0.125 m/step \times 32 steps = 4 m\). Calculate the total time taken to cover this distance by dividing the total height by the speed of the escalator: \(4 m / 0.6 m/s = 6.67 s\).
03

Calculate Work Done

The work done in lifting mass in the gravitational field (which effectively is the gravitational potential energy) is calculated by multiplying the mass, gravitation field strength, and height, i.e, \(150 kg \times 9.81 m/s^2 \times 4 m = 5886 J\), where \(9.81 m/s^2\) is the approximate value of gravity on Earth, \(150 kg\) is the mass, and \(4 m\) is the total height.
04

Calculate Power

Finally, calculate the power, which is the work done per unit time. So, power = work / time = \(5886 J / 6.67 s = 882 W\) (watts).

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Most popular questions from this chapter

A \(60-\mathrm{kg}\) satellite travels in free flight along an elliptical orbit such that at \(A\), where \(r_{A}=20 \mathrm{Mm}\), it has a speed \(v_{A}=40 \mathrm{Mm} / \mathrm{h}\). What is the speed of the satellite when it reaches point \(B\), where \(r_{B}=80 \mathrm{Mm}\) ? Hint: See Prob. \(\quad 14-82, \quad\) where \(\quad M_{e}=5.976\left(10^{24}\right) \mathrm{kg} \quad\) and \(G=66.73\left(10^{-12}\right) \mathrm{m}^{3} /\left(\mathrm{kg} \cdot \mathrm{s}^{2}\right)\)

The "flying car" is a ride at an amusement park which consists of a car having wheels that roll along a track mounted inside a rotating drum. By design the car cannot fall off the track, however motion of the car is developed by applying the car's brake, thereby gripping the car to the track and allowing it to move with a constant speed of the track, \(v_{t}=3 \mathrm{~m} / \mathrm{s}\). If the rider applies the brake when going from \(B\) to \(A\) and then releases it at the top of the drum, \(A\), so that the car coasts freely down along the track to \(B(\theta=\pi \mathrm{rad})\), determine the speed of the car at \(B\) and the normal reaction which the drum exerts on the car at \(B\). Neglect friction during the motion from \(A\) to \(B .\) The rider and car have a total mass of \(250 \mathrm{~kg}\) and the center of mass of the car and rider moves along a circular path having a radius of \(8 \mathrm{~m}\).

The sports car has a mass of \(2.3 \mathrm{Mg}\), and while it is traveling at \(28 \mathrm{~m} / \mathrm{s}\) the driver causes it to accelerate at \(5 \mathrm{~m} / \mathrm{s}^{2} .\) If the drag resistance on the car due to the wind is \(F_{D}=\left(0.3 v^{2}\right) \mathrm{N}\), where \(v\) is the velocity in \(\mathrm{m} / \mathrm{s}\), determine the power supplied to the engine at this instant. The engine has a running efficiency of \(\varepsilon=0.68\).

Determine the power input for a motor necessary to lift \(300 \mathrm{lb}\) at a constant rate of \(5 \mathrm{ft} / \mathrm{s}\). The efficiency of the motor is \(\varepsilon=0.65\)

The 1000 -lb elevator is hoisted by the pulley system and motor \(M\). If the motor exerts a constant force of \(500 \mathrm{lb}\) on the cable, determine the power that must be supplied to the motor at the instant the load has been hoisted \(s=15 \mathrm{ft}\) starting from rest. The motor has an efficiency of \(\varepsilon=0.65\).
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