Ch 10: Interactions and Potential 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 10: Interactions and Potential Energy

Problem 14a

Chapter 10, Problem 14a

In a hydroelectric dam, water falls 25 m and then spins a turbine to generate electricity. What is $Δ U_{G}$ of 1.0 kg of water?

Verified step by step guidance

Step 1: Understand the concept of gravitational potential energy (Ug). Gravitational potential energy is given by the formula:

U g = m g h

, where

m

is the mass of the object,

g

is the acceleration due to gravity, and

h

is the height.

Step 2: Identify the given values in the problem. The mass of the water is

1.0

kg, the height is

25

m, and the acceleration due to gravity is approximately

9.8

m/s².

Step 3: Substitute the given values into the formula for gravitational potential energy. The change in gravitational potential energy,

∆ U g

, is calculated as:

∆ U g = m g h

Step 4: Perform the substitution step explicitly. Replace

m

with

1.0

kg,

g

with

9.8

m/s², and

h

with

25

Step 5: Multiply the values together to find the change in gravitational potential energy. The result will be in joules (J), as gravitational potential energy is measured in joules.

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Key Concepts

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

Gravitational Potential Energy (Ug)

Gravitational potential energy (Ug) is the energy an object possesses due to its position in a gravitational field. It is calculated using the formula Ug = mgh, where m is the mass, g is the acceleration due to gravity (approximately 9.81 m/s²), and h is the height above a reference point. In the context of the hydroelectric dam, the potential energy of the water decreases as it falls, which can be quantified to determine the energy available for conversion into electricity.

Energy Conservation

The principle of energy conservation states that energy cannot be created or destroyed, only transformed from one form to another. In a hydroelectric dam, the gravitational potential energy of the falling water is converted into kinetic energy as it moves downward, and then into mechanical energy as it spins the turbine. This transformation is crucial for understanding how the dam generates electricity from the energy stored in the water.

Work Done by Gravity

The work done by gravity on an object is the energy transferred to or from the object as it moves in the gravitational field. For the falling water in the dam, the work done by gravity can be calculated as the change in gravitational potential energy as the water descends. This work is what ultimately drives the turbine, making it essential to calculate the energy change to determine how much energy is available for electricity generation.

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Work Done By Gravity

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The maximum energy a bone can absorb without breaking is surprisingly small. Experimental data show that a leg bone of a healthy, 60 kg human can absorb about 200 J. From what maximum height could a 60 kg person jump and land rigidly upright on both feet without breaking his legs? Assume that all energy is absorbed by the leg bones in a rigid landing.

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Textbook Question

In a hydroelectric dam, water falls 25 m and then spins a turbine to generate electricity. Suppose the dam is 80% efficient at converting the water's potential energy to electrical energy. How many kilograms of water must pass through the turbines each second to generate 50 MW of electricity? This is a typical value for a small hydroelectric dam.

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Textbook Question

A stretched spring stores 2.0 J of energy. How much energy will be stored if the spring is stretched three times as far?

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In a hydroelectric dam, water falls 25 m and then spins a turbine to generate electricity. What is ΔUG\(\Delta\) U_{G} of 1.0 kg of water?

Verified video answer for a similar problem:

Key Concepts

Gravitational Potential Energy (Ug)

Energy Conservation

Work Done by Gravity

In a hydroelectric dam, water falls 25 m and then spins a turbine to generate electricity. What is $Δ U_{G}$ of 1.0 kg of water?