Problem 12 - Applied Physics

Question

The Hoover Dam is \(726 \mathrm{ft}\) high. Find the potential energy of \(1.00\) million \(\mathrm{ft}^{3}\) of water at the top of the dam. (1 \(\mathrm{ft}^{3}\) of water weighs \(62.4 \mathrm{lb}\).)

Step-by-Step Solution

Answer

The potential energy is 1,460,563,200,000 foot-pounds.

Step 1: Understand the Problem

We want to find the potential energy of 1 million cubic feet of water at the top of a dam that is 726 feet tall. The potential energy can be found using the formula \( PE = mgh \), where \( m \) is the mass (weight, since we're using pounds), \( g \) is the acceleration due to gravity (in feet per second squared), and \( h \) is the height.

Step 2: Calculate the Total Weight of Water

We know that 1 cubic foot of water weighs 62.4 pounds. Therefore, 1 million cubic feet of water weighs:\[\text{Total Weight} = 1,000,000 \, \text{ft}^3 \times 62.4 \, \text{lb/ft}^3 = 62,400,000 \, \text{lb}\]

Step 3: Use Potential Energy Formula

Use the potential energy formula \( PE = mgh \), where \( m = 62,400,000 \, \text{lb} \), \( g = 32.2 \, \text{ft/s}^2 \), and \( h = 726 \, \text{ft} \). Therefore, the potential energy is:\[PE = 62,400,000 \, \text{lb} \times 32.2 \, \text{ft/s}^2 \times 726 \, \text{ft}\]

Step 4: Calculate Potential Energy

Now, compute the potential energy:\[PE = 62,400,000 \times 32.2 \times 726 = 1,460,563,200,000 \, \text{ft} \cdot \text{lb}\]Thus, the potential energy of 1 million cubic feet of water at the top of the dam is 1,460,563,200,000 foot-pounds.

Key Concepts

Hoover Dam

The Hoover Dam is an iconic structure located on the border between Nevada and Arizona. It is admired not just for its architectural significance but also its sheer size and the role it plays in harnessing water for energy. Standing at 726 feet tall, the dam is one of the most massive concrete structures in the world, holding back the Colorado River to form Lake Mead. This height is crucial when computing gravitational potential energy as it directly influences the amount of energy stored due to the water's elevated position. At the top of the dam, the water's potential energy is at its maximum. Understanding these physical properties helps in solving physics problems related to energy and forces, particularly those involving large-scale hydroelectric facilities like the Hoover Dam.

Physics Problem Solving

Physics problem-solving often requires breaking down complex systems into manageable parts. The problem of finding the potential energy of water at the top of the Hoover Dam uses this technique.

• First, understand the problem - identify what you need to find and what is given.
• Second, apply relevant formulas - for potential energy, the formula is typically expressed as \( PE = mgh \).
• Next, substitute known values into the formula - be attentive to units and conversion factors.
• Finally, calculate the answer, double-checking your math to ensure accuracy.

By practicing this structured approach, you'll sharpen your problem-solving skills in physics, enabling you to tackle various scenarios confidently.

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses because of its position in a gravitational field. The formula \( PE = mgh \) is used where \( m \) stands for mass, \( g \) represents the gravitational acceleration, and \( h \) is the height above a reference point. When water is stored at the top of the Hoover Dam, its potential energy increases with its height. The gravity constantly pulls it downward. When water is released from the dam, this stored energy can be converted into kinetic energy, utilized for generating electricity in a hydroelectric plant. This transformation makes gravitational potential energy a crucial concept in renewable energy technologies. It's important to remember that this energy is directly proportional to the mass of water and the height, as illustrated by the exercise.

Mass and Weight Calculation

Calculating mass and weight is a fundamental skill in physics and is especially crucial in solving problems involving potential energy. In this example, the weight of the water is determined by the volume and its density, converting volume into weight using the provided density. For the Hoover Dam problem:

• Each cubic foot of water weighs 62.4 pounds.
• To find the total weight, multiply the volume of water by its weight per cubic unit: \( 1,000,000 \, \text{ft}^3 \times 62.4 \, \text{lb/ft}^3 = 62,400,000 \, \text{lb} \).

This calculation forms the basis for determining potential energy since the formula \( PE = mgh \) requires the mass in weight form. Always double-check these calculations to ensure consistency and accuracy, as they are critical for precise energy estimations.