/*! 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 44 An automobile having a mass of \... [FREE SOLUTION] | 91影视

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Chapter 14: Problem 44

An automobile having a mass of \(2 \mathrm{Mg}\) travels up a \(7^{\circ}\) slope at a constant speed of \(v=100 \mathrm{km} / \mathrm{h}\). If mechanical friction and wind resistance are neglected, determine the power developed by the engine if the automobile has an efficiency \(\varepsilon=0.65\)

Short Answer

Expert verified
The amount of power developed by the car's engine to move up the slope while maintaining the given speed and efficiency is approximately 1021 kW.

Step by step solution

01

Calculate the force exerted due to gravity

Firstly, compute the gravitational force acting on the automobile. For an object on an incline, the force due to gravity is \(m \cdot g \cdot \sin(\Theta)\), where \(m\) is the mass, \(g\) is acceleration due to gravity, and \(\Theta\) is the angle of slope. For this problem the mass \(m\) of the car is \(2 \, \text{Mg} = 2000 \, \text{kg}\), gravity \(g\) is \(9.8 \, \text{m/s}^2\), and the angle of slope \(\Theta\) is \(7^\circ\). Taking \(\sin(7^\circ) \approx 0.122\), we can calculate the gravitational force: \(F_g = m \cdot g \cdot \sin(\Theta) = 2000 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 \cdot 0.122 \approx 2.39 \times 10^4 \, \text{N}\).
02

Determine the work done by the engine

Considering the conditions of constant speed and neglecting mechanical friction and wind resistance, the engine has to overcome exactly the gravitation force. Hence, the work done \(W\) by the engine for every meter climbed is equal to the gravitational force: \(W = F_g = 2.39 \times 10^4 \, \text{N}\).
03

Compute the Speed in m/s

The speed given in the problem is in km/hr. However, in this context, it's convenient to convert this to m/s. Using the conversion factor 1 km/hr = 0.2778 m/s, we get \( v = 100 \, \text{km/hr} \cdot 0.2778 = 27.78 \, \text{m/s}\).
04

Compute the Raw Power

Power \(P_r\) is the rate of doing work or energy transfer. So, the raw power required to maintain the speed v against the force \(F_g\) can be calculated using the formula \(P_r = F_g \cdot v = 2.39 \times 10^4 \, \text{N} \cdot 27.78 \, \text{m/s} \approx 6.64 \times 10^5 \, \text{watts} \, \text{or} \, \approx 664 \, \text{kw}\).
05

Account for the Efficiency

The power calculated in the previous step is the raw power. However, the engine has an efficiency \(\varepsilon\) of 0.65. This means only 65% of power developed is used for useful work. Hence, the actual power \(P_a\) developed by the engine would be given by \(P_a = P_r / \varepsilon = 664 \, \text{kw} / 0.65 \approx 1021 \, \text{kw}\).

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Key Concepts

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

Power calculation
Calculating power is a fundamental aspect of engineering mechanics. In this problem, the power generated by an automobile engine can be understood as the rate at which it does work or converts energy. Power is calculated by multiplying the force required to move the object, in this case, a car, by its speed. The formula for calculating raw power is given by:
  • \[ P_r = F_g \times v \]
where \( F_g \) is the gravitational force acting along the incline and \( v \) is the velocity of the car.
Using these values, we can determine the raw power needed to maintain the car's motion against gravitational pull. This serves as a basis for analyzing engine requirements and performance.
Force on incline
When an object moves on an incline, determining the force exerted by gravity is crucial. This force, along the slope, influences how much effort is needed from an engine to maintain speed. The force due to gravity on an incline can be calculated using:
  • \[ F_g = m \cdot g \cdot \sin(\Theta) \]
Here, \( m \) represents the mass of the car, \( g \) is the acceleration due to gravity (approximately \( 9.8 \text{ m/s}^2 \) on Earth), and \( \Theta \) is the angle of the slope. This calculation helps in understanding the actual workload imposed by the incline on the vehicle's engine.
Engine efficiency
Engine efficiency is a measure of how well an engine converts input energy into useful work. It is particularly important in assessing the performance of engines in vehicles. Efficiency \( \varepsilon \) is defined as:
  • \[ \varepsilon = \frac{\text{Useful Work Output}}{\text{Total Energy Input}} \]
In our problem, the engine has an efficiency of 0.65 or 65%. This means that only 65% of the raw power generated is converted into useful work. To find the actual power developed by the engine, you account for this efficiency as follows:
  • \[ P_a = \frac{P_r}{\varepsilon} \]
Understanding efficiency helps in calculating how much additional power is needed for real-world scenarios beyond ideal conditions.
Work and energy
Work and energy are foundational concepts in physics and engineering that describe how forces cause displacement when energy is transferred. Work done by a force is defined as the product of that force and the distance over which it acts. If a car moves up an incline, the work done by its engine to counteract gravity is crucial for maintaining its speed.
  • \[ W = F_g \times d \]
Here, \( W \) is the work, \( F_g \) is the force due to gravity along the incline, and \( d \) is the displacement. Understanding these concepts allows engineers to design systems that effectively use energy, ensuring that enough work is done to overcome environmental challenges, such as driving up a hill.

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Most popular questions from this chapter

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 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 \(\varepsilon=0.65\)

The roller coaster car has a mass of \(700 \mathrm{kg}\) including its passenger. If it starts from the top of the hill \(A\) with a speed \(v_{A}=3 \mathrm{m} / \mathrm{s},\) determine the minimum height \(h\) of the hill crest so that the car travels around the inside loops without leaving the track. Neglect friction, the mass of the wheels, and the size of the car. What is the normal reaction on the car when the car is at \(B\) and when it is at \(C ?\) Take \(\rho_{B}=7.5 \mathrm{m}\) and \(\rho_{C}=5 \mathrm{m}\)

The force of \(F=50 \mathrm{N}\) is applied to the cord when \(s=2 \mathrm{m} .\) If the 6 -kg collar is orginally at rest, determine its velocity at \(s=0 .\) Neglect friction.

The sports car has a mass of \(2.3 \mathrm{Mg}\) and accelerates at \(6 \mathrm{m} / \mathrm{s}^{2},\) starting from rest. If the drag resistance on the car due to the wind is \(F_{D}=(10 v) \mathrm{N},\) where \(v\) is the velocity in \(\mathrm{m} / \mathrm{s},\) determine the power supplied to the engine when \(t=5 \mathrm{s}\). The engine has a running efficiency of \(\varepsilon=0.68\)
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