91Ó°ÊÓ

A small experimental vehicle has a total mass \(m\) of \(500 \mathrm{kg}\) including wheels and driver. Each of the four wheels has a mass of \(40 \mathrm{kg}\) and a centroidal radius of gyration of \(400 \mathrm{mm} .\) Total frictional resistance \(R\) to motion is \(400 \mathrm{N}\) and is measured by towing the vehicle at a constant speed on a level road with engine disengaged. Determine the power output of the engine for a speed of \(72 \mathrm{km} / \mathrm{h}\) up the 10-percent grade (a) with zero acceleration and (b) with an acceleration of \(3 \mathrm{m} / \mathrm{s}^{2}\). (Hint: Power equals the time rate of increase of the total energy of the vehicle plus the rate at which frictional work is overcome.)

Step by step solution

01

Convert Speed to SI Units

First, convert the given speed of the vehicle from kilometers per hour to meters per second. \[ 72 \text{ km/h} = 72 \times \frac{1000}{3600} \text{ m/s} = 20 \text{ m/s} \]

02

Calculate Gravitational Force on the Slope

Determine the component of gravitational force acting along the slope. The grade is 10%, which means the slope angle \( \theta \) satisfies \( \tan(\theta) = 0.1 \). The gravitational force \( F_g \) is calculated as:\[ F_g = mg \sin(\theta) = 500 \times 9.81 \times \frac{0.1}{\sqrt{1^2 + 0.1^2}} \approx 500 \times 9.81 \times 0.1 = 490.5 \text{ N} \]

03

Calculate Power for Zero Acceleration

With zero acceleration, the rate of increase of total energy is zero, and power must overcome frictional resistance and gravitational force:\[ P_0 = (R + F_g) \times v = (400 + 490.5) \times 20 = 17810 \text{ W} = 17.81 \text{ kW} \]

04

Calculate Inertial Force Due to Acceleration

Calculate the inertial force required to accelerate the vehicle with \( a = 3 \text{ m/s}^2 \). Four wheels need additional inertial force due to their rotation. First, determine the moment of inertia \( I \) for each wheel:\[ I = mr^2 = 40 \times (0.4)^2 = 6.4 \text{ kg} \cdot \text{m}^2 \]Then calculate the inertial force:\[ F_i = ma + 4 \cdot \left(\frac{I}{r^2}\right) a = 500 \times 3 + 4 \times \left(\frac{6.4}{(0.4)^2}\right) \times 3 = 1500 + 4 \times 40 \times 3 = 1500 + 480 = 1980 \text{ N} \]

05

Calculate Power for Accelerated Motion

For the vehicle with acceleration, the power must also overcome the inertial force:\[ P_a = (R + F_g + F_i) \times v = (400 + 490.5 + 1980) \times 20 = 57410 \text{ W} = 57.41 \text{ kW} \]

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Key Concepts

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

Power Output Calculation

Power output is crucial in determining how much energy a vehicle's engine is using to maintain a given performance level. It is especially important when overcoming various resistances such as friction and gravity, or when accelerating a vehicle. Power, measured in watts (W), is the rate at which work is done or energy is transferred. To calculate the power output for a vehicle:

• We need to sum up all the resistive forces, including frictional and gravitational forces.
• Calculate the force required for any acceleration if the vehicle is speeding up.
• Use the formula \( P = F \times v \), where \( P \) is power, \( F \) is total force, and \( v \) is velocity in meters per second.

By understanding and calculating power output, you can ensure a vehicle's engine runs efficiently, combining all forces affecting motion for smooth operation.

Frictional Resistance

Frictional resistance is one of the key forces that a vehicle must overcome to maintain motion. It arises from the contact between the vehicle's wheels and the road surface. As the vehicle moves, friction works against the direction of motion, creating a force that the engine needs to overcome. In this exercise:

• The frictional resistance is given as 400 N.
• It's measured by towing the vehicle at a constant speed with the engine turned off.
• This reflects the amount of force needed to maintain motion without accelerating.

Understanding frictional resistance is essential for energy efficiency and optimizing fuel consumption in vehicles.

Gravitational Force

Gravitational force affects vehicles significantly when they travel uphill or downhill. On a slope, gravity works against the vehicle by pulling it back down the incline. This requires additional power from the engine to maintain speed. To calculate this force:

• Use the formula \( F_g = mg \sin(\theta) \), with \( m \) being the vehicle's mass, \( g \) as the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)), and \( \theta \) as the angle of the slope.
• In this exercise, the angle corresponds to a 10% grade, indicating that for every 10 units horizontally, there's a 1-unit rise vertically.

The gravitational force component becomes a critical factor when calculating necessary power outputs on hilly terrains.

Inertial Force

Inertial force comes into play when a vehicle accelerates. This force is what the vehicle needs to overcome to initiate movement or increase speed. It's determined by Newton's second law, \( F = ma \), where \( m \) is mass and \( a \) is acceleration. In this scenario, the vehicle's acceleration is 3 \( \text{m/s}^2 \). However, an additional rotational inertial force from the wheels must be considered, as they have mass and rotate:

• The moment of inertia \( I \) for a wheel is \( mr^2 \).
• The total inertial force includes the linear component and the rotational component due to the wheels.

Accurately calculating inertial force helps in understanding how much effort and energy are necessary to change a vehicle's speed.

Vehicle Dynamics

Vehicle dynamics encompasses all the forces and motions acting on a vehicle, allowing it to move smoothly and efficiently within various conditions. It involves understanding how different forces like friction, gravity, and inertia affect a vehicle's motion:

• Forces such as friction and gravitational effects need to be overcome for motion stability and speed.
• The vehicle's dynamics change according to its speed, road incline, and required acceleration.
• By mastering these dynamics, drivers and engineers can predict performance and design vehicles to meet specific operational needs, ensuring safety and efficiency.

Vehicle dynamics is a comprehensive subject, blending mechanical principles with real-world driving conditions.