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 61

Chapter 9, Problem 61

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?

Verified step by step guidance

Step 1: Start by understanding the relationship between kinetic energy and electric power. The kinetic energy of the water is given by the formula:

E_{k} = (1/2) * m * v

², where

m

is the mass flow rate (kg/s) and

v

is the velocity of the water (m/s).

Step 2: Recognize that the efficiency of the hydroelectric plant is 80%, meaning only 80% of the kinetic energy is converted into electric energy. Therefore, the electric power output

P

is related to the kinetic energy by the equation:

P = 0.8 * (1/2) * m * v

².

Step 3: Rearrange the equation to solve for the mass flow rate

m

. The formula becomes:

m = (2 * P) / (0.8 * v

²).

Step 4: Substitute the given values into the formula. The electric power output

P

is 50 MW (50,000,000 W), and the velocity

v

is 18 m/s. Plug these values into the equation:

m = (2 * 50,000,000) / (0.8 * 18

²).

Step 5: Perform the calculations to find the mass flow rate

m

. This will give you the water flow rate in kilograms per second (kg/s).

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Key Concepts

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

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion, calculated using the formula KE = 1/2 mv², where m is mass and v is velocity. In the context of hydroelectric power, the kinetic energy of falling water is converted into electrical energy by turbines. Understanding this concept is crucial for determining how much energy can be harnessed from the moving water.

Power and Efficiency

Power is the rate at which energy is transferred or converted, measured in watts (W). In this scenario, the efficiency of the hydroelectric plant indicates that 80% of the kinetic energy from the water is converted into electrical energy. This efficiency factor is essential for calculating the actual energy output and understanding how much of the input energy is effectively utilized.

Flow Rate

Flow rate refers to the volume of fluid that passes through a given surface per unit time, often expressed in kilograms per second (kg/s) for water. In hydroelectric plants, the flow rate is critical for determining how much water is needed to generate a specific amount of power. By relating flow rate to the kinetic energy and power output, one can calculate the necessary water flow to achieve the desired electrical generation.

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Related Practice

Textbook Question

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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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. 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?

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Textbook Question

A horizontal spring with spring constant 250 N/m is compressed by 12 cm and then used to launch a 250 g box across the floor. The coefficient of kinetic friction between the box and the floor is 0.23. What is the box's launch speed?

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Textbook Question

A 90 kg firefighter needs to climb the stairs of a 20-m-tall building while carrying a 40 kg backpack filled with gear. How much power does he need to reach the top in 55 s?

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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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