Ch 09: Work and Kinetic Energy

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 09: Work and Kinetic Energy

Problem 62b

Chapter 9, Problem 62b

When you ride a bicycle at constant speed, nearly all the energy you expend goes into the work you do against the drag force of the air. Model a cyclist as having cross-section area 0.45 m² and, because the human body is not aerodynamically shaped, a drag coefficient of 0.90. Use 1.2 kg/m³ as the density of air at room temperature. Metabolic power is the rate at which your body 'burns' fuel to power your activities. For many activities, your body is roughly 25% efficient at converting the chemical energy of food into mechanical energy. What is the cyclist's metabolic power while cycling at 7.3 m/s?

Verified step by step guidance

Step 1: Start by calculating the drag force experienced by the cyclist using the drag force formula: F_d = (1/2) * ρ * C_d * A * v², where ρ is the air density (1.2 kg/m³), C_d is the drag coefficient (0.90), A is the cross-sectional area (0.45 m²), and v is the velocity (7.3 m/s).

Step 2: Once you have the drag force F_d, calculate the mechanical power required to overcome this drag force. Mechanical power is given by P_mech = F_d * v, where F_d is the drag force and v is the velocity (7.3 m/s).

Step 3: Recognize that the cyclist's body is only 25% efficient at converting chemical energy into mechanical energy. To find the total metabolic power, divide the mechanical power by the efficiency: P_metabolic = P_mech / efficiency, where efficiency is 0.25 (25%).

Step 4: Substitute the values for F_d, v, and efficiency into the formula for P_metabolic to calculate the metabolic power. Ensure all units are consistent throughout the calculation.

Step 5: Verify your calculations and ensure that the final metabolic power value is reasonable given the context of the problem. This will give you the cyclist's metabolic power while cycling at 7.3 m/s.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Drag Force

The drag force is the resistance experienced by an object moving through a fluid, such as air. It is influenced by the object's speed, cross-sectional area, and shape, as well as the fluid's density. The drag force can be calculated using the equation F_d = 0.5 * C_d * A * ρ * v², where C_d is the drag coefficient, A is the cross-sectional area, ρ is the fluid density, and v is the velocity. Understanding drag is crucial for analyzing the energy expenditure of a cyclist.

Metabolic Power

Metabolic power refers to the rate at which the body converts chemical energy from food into mechanical energy for physical activities. It is typically measured in watts and is influenced by the efficiency of the body's energy conversion processes. In this context, the efficiency of the human body is about 25%, meaning that only a quarter of the energy consumed is converted into useful work. This concept is essential for determining how much energy a cyclist needs to maintain a certain speed against drag.

Efficiency in Energy Conversion

Efficiency in energy conversion is a measure of how effectively a system converts input energy into useful output energy. In the case of human metabolism, the efficiency is around 25%, indicating that a significant portion of energy is lost as heat rather than being used for mechanical work. This concept is important for calculating the total metabolic power required for a cyclist, as it directly affects the amount of energy that must be consumed to sustain a given speed.

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A spring of equilibrium length L₁ and spring constant k₁ hangs from the ceiling. Mass m₁ is suspended from its lower end. Then a second spring, with equilibrium length L₂ and spring constant k₂, is hung from the bottom of m₁. Mass m₂ is suspended from this second spring. How far is m₂ below the ceiling?

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Textbook Question

When you ride a bicycle at constant speed, nearly all the energy you expend goes into the work you do against the drag force of the air. Model a cyclist as having cross-section area 0.45 m² and, because the human body is not aerodynamically shaped, a drag coefficient of 0.90. Use 1.2 kg/m³ as the density of air at room temperature. The food calorie is equivalent to 4190 J. How many calories does the cyclist burn if he rides over level ground at 7.3 m/s for 1 h?

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A Porsche 944 Turbo has a rated engine power of 217 hp. 30% of the power is lost in the engine and the drive train, and 70% reaches the wheels. The total mass of the car and driver is 1480 kg, and two-thirds of the weight is over the drive wheels. If the Porsche accelerates at a_max, what is its speed when it reaches maximum power output?

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A hydroelectric power plant uses spinning turbines to transform the kinetic energy of moving water into electric energy with 80% efficiency. That is, 80% of the kinetic energy becomes electric energy. A small hydroelectric plant at the base of a dam generates 50 MW of electric power when the falling water has a speed of 18 m/s. What is the water flow rate - kilograms of water per second - through the turbines?

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Textbook Question

A Porsche 944 Turbo has a rated engine power of 217 hp. 30% of the power is lost in the engine and the drive train, and 70% reaches the wheels. The total mass of the car and driver is 1480 kg, and two-thirds of the weight is over the drive wheels. What is the maximum acceleration of the Porsche on a concrete surface where μ_s = 1.00? Hint: What force pushes the car forward?

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