Ch. 08 - Conservation of Energy

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 08 - Conservation of Energy

Problem 79b

Chapter 8, Problem 79b

Water flows slowly over a dam at the rate of 320 kg/s and falls vertically 88 m before striking the turbine blades. Calculate the rate at which mechanical energy is transferred to the turbine blades, assuming 55% efficiency.

Verified step by step guidance

Step 1: Understand the problem. The water falling over the dam has gravitational potential energy, which is converted into mechanical energy as it falls. The rate of energy transfer to the turbine blades depends on the efficiency of the system (55%). We need to calculate the rate of mechanical energy transfer.

Step 2: Write the formula for gravitational potential energy. The potential energy of the water is given by: \( E_p = m g h \), where \( m \) is the mass of the water (in kg), \( g \) is the acceleration due to gravity (\( 9.8 \ \text{m/s}^2 \)), and \( h \) is the height (in meters).

Step 3: Since the problem involves a rate (320 kg/s), calculate the power associated with the gravitational potential energy. Power is the rate of energy transfer, so: \( P = \dot{m} g h \), where \( \dot{m} \) is the mass flow rate (320 kg/s). Substitute the known values for \( \dot{m} \), \( g \), and \( h \).

Step 4: Account for the efficiency of the turbine. Only 55% of the gravitational potential energy is converted into mechanical energy. Multiply the power calculated in Step 3 by the efficiency (0.55) to find the rate of mechanical energy transfer: \( P_{\text{mechanical}} = P \times 0.55 \).

Step 5: Simplify the expression to find the final result. Ensure all units are consistent (e.g., power in watts) and verify the calculation. The result will give the rate at which mechanical energy is transferred to the turbine blades.

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Key Concepts

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

Mechanical Energy

Mechanical energy is the sum of potential and kinetic energy in a system. In the context of the water flowing over the dam, the potential energy is due to its height (88 m), which can be converted into kinetic energy as it falls. This energy is then transferred to the turbine blades, where it is converted into mechanical energy to do work.

Efficiency

Efficiency is a measure of how much input energy is converted into useful output energy. In this scenario, the efficiency of the turbine is given as 55%, meaning that only 55% of the mechanical energy generated from the falling water is effectively used to perform work. The remaining energy is lost, often as heat or sound.

Rate of Energy Transfer

The rate of energy transfer refers to how quickly energy is converted from one form to another, typically measured in watts (Joules per second). In this case, it involves calculating the mechanical energy transferred to the turbine blades per second based on the mass flow rate of the water and the height from which it falls, adjusted for the turbine's efficiency.

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The graph of Fig. 8–43 shows the potential energy curve of a particle moving along the 𝓍 axis under the influence of a conservative force. Note that the total energy E > U(𝓍), so that the particle’s speed is never zero. In which interval(s) of 𝓍 is the force on the particle to the right?

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Textbook Question

The graph of Fig. 8–43 shows the potential energy curve of a particle moving along the 𝓍 axis under the influence of a conservative force. Note that the total energy E > U(𝓍), so that the particle’s speed is never zero. At what value(s) of 𝓍 is the magnitude of the force a minimum?

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