Calculating the Work Done by a Crane Lifting a Steel Beam: A Comprehensive Tutorial

A crane lifting a 425 kg steel beam 95 m upward with an acceleration of 1.8 m/s2 presents an interesting problem in physics. This article explains the process to calculate the work done by the crane using both direct and advanced methods. By understanding these calculations, we can better appreciate the mechanical and physical principles at play.

Introduction to the Problem

The crane not only needs to lift the beam against gravity but also to provide the necessary force to accelerate the beam upward. This requires a combination of gravitational force and additional force for acceleration. Let's break down the problem into steps and solve it.

Step-by-Step Calculation of Work Done

Step 1: Calculate the Gravitational Force Acting on the Beam

The gravitational force (weight) can be calculated using the formula:

F_{gravity} m cdot g

where:

m 425 kg (mass of the beam) g 9.81 m/s2 (acceleration due to gravity)

Let's calculate the gravitational force:

F_{gravity} 425 ; text{kg} cdot 9.81 ; text{m/s}^2 4162.25 ; text{N}

Step 2: Calculate the Net Force Required to Accelerate the Beam

The net force needed to accelerate the beam upward can be calculated using Newton’s second law:

F_{net} m cdot a

where:

a 1.8 m/s2 (upward acceleration)

Let's calculate the net force:

F_{net} 425 ; text{kg} cdot 1.8 ; text{m/s}^2 765 ; text{N}

Step 3: Calculate the Total Force Exerted by the Crane

The total force exerted by the crane must overcome both the gravitational force and provide the net force for the upward acceleration:

F_{total} F_{gravity} F_{net}

Plugging in the values calculated in steps 1 and 2:

begin{aligned}F_{total} 4162.25 ; text{N} 765 ; text{N} 4927.25 ; text{N}end{aligned}

Step 4: Calculate the Work Done by the Crane

The work done by the crane can be calculated using the formula:

W F cdot d

where:

F F_{total} 4927.25 N (total force exerted by the crane) d 95 m (distance lifted)

Let's calculate the work done:

W 4927.25 ; text{N} cdot 95 ; text{m} 467086.75 ; text{J}

Conclusion: The work done by the crane on the beam is approximately 467087 Joules (to three significant figures).

Alternative Approach to Solve the Problem

Akin to understanding the first approach, this problem can also be solved by considering the changes in the kinematic parameters, specifically velocity and displacement. We can calculate the final velocity v_f to determine the increase in kinetic energy and the change in potential energy.

Step 1: Calculate the Final Velocity

We know that the beam is accelerating at 1.8 m/s2 over a distance of 95 m. The starting velocity is assumed to be 0 m/s. Using the equation for final velocity:

v_f^2 v_i^2 2a Delta x

a 1.8 m/s2 Δx 95 m v_i 0 m/s

Solving for v_f:

v_f^2 0 ; text{m/s}^2 2 cdot 1.8 ; text{m/s}^2 cdot 95 ; text{m} 342 ; text{m}^2/text{s}^2

Therefore, v_f sqrt{342} approx 18.5 ; text{m/s}

Step 2: Calculate the Work Done Using Kinetic and Potential Energy

The work done by the crane can be expressed as the difference in kinetic and potential energy:

W Delta K Delta U

Where:

Delta K m(frac{v_f^2 - v_i^2}{2}) Delta U mgh

Plugging in the values:

begin{aligned}W 425 ; text{kg} cdot left(frac{18.5 ; text{m/s}^2 - 0 ; text{m/s}^2}{2}right) 425 ; text{kg} cdot 9.81 ; text{m/s}^2 cdot 95 ; text{m} 425 ; text{kg} cdot 171 ; text{m}^2 / text{s}^2 425 ; text{kg} cdot 932 ; text{m}^2 / text{s}^2 73275 ; text{J} 394300 ; text{J} 468575 ; text{J}end{aligned}

Conclusion: The work done by the crane using the alternative method is approximately 469000 Joules, which is in close agreement with the initial calculation.

Conclusion

This detailed calculation demonstrates the physics behind the work done by a crane lifting a heavy object. Understanding both the direct method and the energy-based approach provides a comprehensive understanding of how forces and energy contribute to the lifting process. Utilizing these calculations is crucial in various engineering and construction applications.