Question

Calculate the total potential energy, in \(\mathrm{Btu},\) of an object with a mass of 200 lbm when it is \(10 \mathrm{ft}\) above a datum level at a location where standard gravitational acceleration exists

Short answer

Question: Calculate the total potential energy in British Thermal Units (Btu) of an object with a mass of 200 pounds-mass (lbm) and a height of 10 feet above the datum level. Answer: The total potential energy of the object is approximately 2.572 Btu.

Step-by-step solution

Convert mass from lbm to slugs

Divide the mass by the conversion factor (32.2 lbm/slug) to obtain the mass in slugs: \(m_{slug} = \dfrac{m_{lbm}}{32.2}\)

Calculate the potential energy in ft-lbf

Use the formula for potential energy: \(PE = m_{slug}gh\) Where \(g\) is the standard gravitational acceleration (32.2 ft/s²) and \(h\) is the height in feet.

Convert potential energy from ft-lbf to Btu

Multiply the potential energy in ft-lbf by the conversion factor: \(PE_{Btu} = PE_{ft-lbf} \times \dfrac{1 \space Btu}{778.16 \space ft-lbf}\) Now, we will apply these steps to the given problem. #Solution#

Convert mass from lbm to slugs

We are given the mass of the object as 200 lbm. We will convert it to slugs using the conversion factor (32.2 lbm/slug): \(m_{slug} = \dfrac{200 \space lbm}{32.2} = 6.211 \space slugs\)

Calculate the potential energy in ft-lbf

We are given the height of the object as 10 ft. Using the formula for potential energy with \(g = 32.2 \space ft/s²\) and \(h = 10 \space ft\), we have: \(PE_{ft-lbf} = m_{slug}gh = 6.211 \space slugs \times 32.2 \space ft/s² \times 10 \space ft = 2000 \space ft-lbf\)

Convert potential energy from ft-lbf to Btu

Finally, we will convert the potential energy in ft-lbf to Btu using the conversion factor (1 Btu = 778.16 ft-lbf): \(PE_{Btu} = PE_{ft-lbf} \times \dfrac{1 \space Btu}{778.16 \space ft-lbf} = 2000 \space ft-lbf \times \dfrac{1 \space Btu}{778.16 \space ft-lbf} = 2.572 \space Btu\) Therefore, the total potential energy of the object when it is 10 ft above the datum level is approximately 2.572 Btu.

Key concepts

Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy that an object possesses due to its position in a gravitational field. The Earth's gravity exerts a force on objects that is proportional to their mass and the distance they are from the center of the earth, or another defined level. GPE can be calculated with the formula:
\(PE = mgh\).
In this formula, \(m\) represents the mass of the object, \(g\) is the acceleration due to gravity, and \(h\) is the height of the object above the datum level.
When calculating GPE, it is important to use consistent units. In the context of our problem, the mass was initially given in pounds mass (lbm), and the height in feet (ft), with the acceleration due to gravity at standard Earth gravity, which is 32.2 feet per second squared (ft/s²). Therefore, ensuring that the units align is the crux for an accurate GPE calculation.
The power of understanding gravitational potential energy lies in its ability to predict how much energy is released when the object is allowed to fall. This is vital for applications ranging from engineering to energy management.

Unit Conversion

Unit conversion is an essential skill in physics and engineering, as it ensures that all the measurements are in compatible units to perform calculations accurately. In the given exercise, the mass was initially presented in pounds mass (lbm) but needed to be converted to slugs, the standard unit of mass in the English Engineering system for these types of calculations.
\(m_{slug} = \frac{m_{lbm}}{32.2}\)
It is crucial to familiarize oneself with common conversion factors, such as \(1 \text{ slug} = 32.2 \text{ lbm}\) for mass and \(1 \text{ Btu} = 778.16 \text{ ft-lbf}\) for energy. Converting units may sometimes seem tedious, but it is a key step in solving physics problems, ensuring that calculated answers are not only numerically correct but also meaningful in the context of the question. Simple mistakes in unit conversion can result in significant errors, so it's always worth double-checking your work.

Energy in Thermodynamics

Energy in thermodynamics is a fundamental concept that concerns the study of heat, work, and the total energy of systems. The potential energy calculated in this exercise could be viewed in a thermodynamic context when we consider the energy content of systems in terms of heat units.
In thermodynamics, the British Thermal Unit (Btu) is often used to measure heat content or energy. It is defined as the amount of heat required to raise the temperature of one pound of water by one degree Fahrenheit. In our solution, we converted the potential energy from foot-pounds-force (ft-lbf) into Btu to express that energy in a form that is readily applicable within the field of thermodynamics.
h4. Practical Relevance
Grasping the thermodynamic significance of energy conversion from mechanical forms, like potential energy, to heat can be key for various engineering tasks including HVAC system design, understanding engine efficiency, or even calculating the energy requirements for heating a space. The transfer and transformation of energy are what allows us to harness the natural world for human needs, and thus, makes thermodynamics a cornerstone in practical science and technology fields.