Problem 10 - Applied Physics

Question

The potential energy of a girder, after being lifted to the top of a building, is \(5.17 \times 10^{5} \mathrm{ftlb}\). If its mass is 173 slugs, how high is the girder?

Step-by-Step Solution

Answer

The girder is raised to a height of approximately 92.8 feet.

Step 1: Identify the Formula

The potential energy (PE) can be calculated using the formula: \[PE = mgh\]where \(m\) is the mass in slugs, \(g\) is the acceleration due to gravity, and \(h\) is the height in feet. In this problem, we are given \(PE\) and \(m\) and need to solve for \(h\).

Step 2: List Known Values

The potential energy \(PE\) is \(5.17 \times 10^5\) ftlb. The mass \(m\) is 173 slugs. The acceleration due to gravity \(g\) is approximately 32.2 ft/s² for calculations in feet and slugs.

Step 3: Rearrange the Formula

We need to solve for the height \(h\), so rearrange the formula:\[h = \frac{PE}{mg}\]

Step 4: Plug in the Known Values

Substitute the known values into the rearranged formula:\[h = \frac{5.17 \times 10^5}{173 \times 32.2}\]

Step 5: Calculate the Height

Perform the division to find the height:\[h = \frac{5.17 \times 10^5}{5570.6} \approx 92.8 \text{ feet}\]

Step 6: Finalize the Answer

Based on the calculations, the girder is raised to a height of approximately 92.8 feet.

Key Concepts

Physics Problem Solving

When tackling physics problems, the first step is to understand what the problem is asking. In this case, we have a girder lifted to a height, and we need to determine this height using the potential energy provided.
To solve it, gather all the known values and the relationships involved. Here, we are dealing with potential energy, mass, and gravitational force. Identifying the right equation to connect these values is crucial. For potential energy, the formula used is:

• \[PE = mgh\]

In our problem, we know the potential energy (\(PE\)) and the mass (\(m\)). Our task is to solve for the height (\(h\)).
After identifying the formula, rearrange it to solve for the unknown variable. Noticing how variables relate will ease this process. Sometimes it might require rearranging, such as:\[h = \frac{PE}{mg}\] This systematic approach of understanding, identifying relationships, rearranging, and solving is the foundation of physics problem-solving. By doing so, the complex problems become more approachable.

Gravitational Potential Energy

Gravitational potential energy is a form of energy that an object possesses due to its position in a gravitational field. This energy increases with height, meaning the higher an object is above the ground, the more potential energy it holds.
The key components of gravitational potential energy are:

• Mass (\(m\)) of the object
• Acceleration due to gravity (\(g\))
• Height (\(h\)) above the reference point

The equation \[PE = mgh\] clearly shows how each component contributes. For our girder, as it is lifted higher, the gravitational potential energy increases proportionally. Understanding how potential energy is converted to and from other forms, like kinetic energy during free fall, is essential in physics. Gravitational potential energy plays a key role in fields like mechanics, construction engineering, and even amusement park design! It tells us how much work is needed to lift an object and the energy it can release when let to fall.

Units of Measurement in Physics

Units of measurement help us standardize and communicate scientific concepts and calculations effectively. In physics, understanding these units is crucial for solving problems accurately.
For our exercise, we encounter:

• Potential energy measured in foot-pounds (ftlb), a unit used primarily in the US customary system.
• Mass given in slugs, which is particularly used in dynamics and is equal to 32.2 pounds-mass.
• Acceleration due to gravity considered as feet per second squared (ft/s²), approximated as 32.2 ft/s².

These units help to calculate the potential energy across different systems by maintaining consistency in the measurements. Although Units like meters per second squared (m/s²) and kilograms are more common internationally, understanding other systems like the US customary helps bridge knowledge across cultural scientific practices. Understanding how to convert and work with these units is an essential skill in physics, enabling problem-solving across different regions and applications.