Calculating Work Done by a Constant Force: A Practical Example

Understanding the concept of work in physics is essential for comprehending energy transfer and mechanical interactions. In this article, we will dive into a practical example of calculating the work done by a constant force, specifically when the force has components in pounds and the displacement is measured in feet. We will explore the principles of the dot product and how it relates to vector-based calculations in two and three-dimensional space.

Introduction to Work and Force in Physics

Work is defined as the product of a force and the displacement when the force is applied in the direction of the displacement. Mathematically, it is represented as the dot product of the force vector and the displacement vector.

Practical Example: Force and Displacement Vectors

Consider a scenario where a constant force F is acting on a particle. The force is given as F -3i - 4j - k, and the particle moves from point P(2, 5, 0) to point Q(-9, 4, -6) on a straight line. Our goal is to calculate the work done by this force during the displacement of the particle.

Steps to Calculate Work Done

Identify the Force Vector: The force vector is F -3i - 4j - k. Determine the Displacement Vector: The particle moves from point P to point Q. The displacement vector Δr is calculated as the difference between the position vectors of the final point Q and the initial point P. Calculate the Displacement Vector: The position vector of point P is 2i 5j 0k, and the position vector of point Q is -9i 4j - 6k. The displacement vector is:

Step-by-Step Calculation:

Find the Displacement Vector: Δr Q - P (-9i 4j - 6k) - (2i 5j 0k) -11i - 1j - 6k. Calculate the Dot Product: The work done W is calculated as the dot product of the force vector and the displacement vector, which is represented mathematically as: W F · Δr (-3i - 4j - k) · (-11i - j - 6k)

The dot product of these two vectors is calculated as:

Calculate the Inner Product: W (-3)(-11) (-4)(-1) (-1)(-6) 33 4 6 43.

Conclusion

The work done by the force F -3i - 4j - k acting on a particle that moves from point P (2, 5, 0) to point Q (-9, 4, -6) is 43 foot-pounds (ft/lbs).

Understanding the Importance of Work in Physics

The concept of work is fundamental in physics and engineering, as it helps in understanding the energy transfer that occurs when a force is applied to an object, leading to its displacement. This principle is crucial for various applications, including calculating the power generated by machines, analyzing the motion of objects, and designing efficient systems.

Further Reading and Applications

For a deeper understanding of work and its applications in physics and engineering, you can explore the following resources:

Books on Classical Mechanics Online Tutorials on Vector Calculations Engineering Physics Textbooks

Understanding work and its applications can enhance your problem-solving skills and provide a solid foundation for more advanced topics in physics and engineering.