Question

A construction crane lifts a prestressed concrete beam weighing 3 short tons from the ground to the top of piers that are \(36 \mathrm{ft}\) above the ground. Determine the amount of work done considering ( \(a\) ) the beam and ( \(b\) ) the crane as the system. Express your answers in both lbf-ft and Btu.

Short answer

Answer: The work done by the construction crane in both cases (a) and (b) is 216000 lbf-ft (≈ 277.5 Btu).

Step-by-step solution

Calculate the force to lift the beam

First, we need to find the force required to lift the beam. Since the weight of the beam is given in short tons, we need to convert it to pounds (lbs). Weight of the beam = 3 short tons * 2000 lbs/short ton = 6000 lbs The force required to lift the beam is equal to its weight. Force = 6000 lbs

Calculate the work done in lifting the beam for case (a)

In case (a), we only consider the beam as the system. To find the work done by the crane, we will multiply the force by the distance through which it is applied. Work done (lbf-ft) = Force (lbs) * Distance (ft) Work done (lbf-ft) = 6000 lbs * 36 ft Work done (lbf-ft) = 216000 lbf-ft

Convert the work done from lbf-ft to Btu for case (a)

To convert the work done from lbf-ft to Btu, we will use the conversion factor 1 Btu = 778.169 lbf-ft. Work done (Btu) = Work done (lbf-ft) / 778.169 lbf-ft/Btu Work done (Btu) = 216000 lbf-ft / 778.169 lbf-ft/Btu Work done (Btu) ≈ 277.5 Btu

Calculate the work done for case (b)

In case (b), we consider both the beam and the crane as the system. The work done on the crane is zero as it does not change its position or height. Therefore, the total work done in case (b) is the same as that in case (a). Work done (b) (lbf-ft) = 216000 lbf-ft Work done (b) (Btu) ≈ 277.5 Btu

Summarize the results

The work done by the construction crane in lifting the prestressed concrete beam from the ground to the top of piers that are 36 ft above the ground is 216000 lbf-ft (≈ 277.5 Btu) for both cases (a) and (b).

Key concepts

Force Calculation

In the realm of physics and engineering, force calculation is a fundamental concept that governs how objects interact with their environment. To comprehend how a construction crane lifts a heavy beam, we must first understand the concept of force. Force is essentially a push or a pull upon an object resulting from its interaction with another object. In most cases, this force is due to gravity, which acts on all objects with mass and pulls them towards the center of the Earth.

When calculating the force required to lift the concrete beam, we equate the force with the weight of the beam. The weight can be calculated by taking the mass of the object (in this case, the beam's weight in short tons) and multiplying it by the acceleration due to gravity. Once we have the weight in the correct units, which is pounds (lbs) for this problem, we've achieved the force required to lift the beam. This calculation serves as the basis to determine the amount of work done, subsequently aiding in understanding the energy conversion that takes place during the lifting process.

Energy Conversion

Energy conversion is another key concept in thermodynamics, which pertains to the transformation of energy from one form to another. When the crane does work to lift the beam, it's converting electrical or mechanical energy, depending on the crane's power source, into gravitational potential energy of the raised beam.

The work done, as calculated in the exercise solutions, quantifies this energy transformation. By multiplying the force required to lift the beam by the distance it was raised, we identify the amount of mechanical energy converted into potential energy. This energy can also be expressed in different units, such as British thermal units (Btu), which represent the amount of energy required to heat one pound of water by one degree Fahrenheit. The conversion from foot-pounds to Btu involves a standardized conversion factor, reflecting the idea that the same amount of work can be described in varying energy terms depending on the context or preferred unit of measurement.

Thermodynamic Systems

Thermodynamic systems are conceptual constructs used to analyze energy and work within the governed physical bounds. In thermodynamics, a 'system' refers to the portion of the universe that is under consideration, separated by boundaries from its 'surroundings'. These systems can be isolated, closed, or open, depending on the exchange of energy and matter with the surroundings.

In the context of the exercise, the 'system' can be defined in two different ways, altering the interpretation of work done. When we consider only the beam as the system (case a), the work calculated reflects the energy transferred to the beam alone. Conversely, when expanding the system to include the crane (case b), it’s important to recognize that the internal movements of the crane do not amount to work done on the crane-beam system, because the crane's position relative to the Earth remains unchanged. This concept highlights how defining system boundaries is crucial for analyzing work and energy flows within thermodynamic processes.