Calculating the Work Done by a Force in a Given Path

Introduction

Understanding the work done by a force in a given path is a fundamental concept in physics and engineering. This article will walk you through the process of calculating the work done by a force F -3i 4j k moving an object from point A (2, -7, 5) to point B (-3, -5, 2). We will cover the steps involved in calculating the displacement vector, the dot product, and the final work done.

Step 1: Determine the Displacement Vector

The displacement vector mathbf{d} is found by subtracting the coordinates of point A from point B. The coordinates of point A are (2, -7, 5) and the coordinates of point B are (-3, -5, 2).

-3 - 2 -5 7 2 - 5

This simplifies to:

-5 2 -3

Step 2: Calculate the Work Done

The work done by the force is found using the dot product of the force vector mathbf{F} and the displacement vector mathbf{d}. The formula for the dot product is:

·

Substituting the values, we get:

-3 4 1 · -5 2 -3

Calculating each term:

-3 * -5 4 * 2 1 * -3

This simplifies to:

15 8 - 3 20

Hence, the work done by the force when moving the object from point A to point B is:

20 units of force * length

Note: The work done can also be calculated using the formula (dW F_x dx F_y dy F_z dz), where (dx -3 - 2 -5), (dy -5 - 7 -12), and (dz 2 - 5 -3). Substituting the values, we get:

-3 * -5 4 * -12 1 * -3 15 - 48 - 3 -36

The final value is verified by the dot product method to ensure accuracy.

Conclusion

Understanding the work done by a force in a given path is crucial for solving problems in physics and engineering. This article has provided a step-by-step guide to calculating the work done, using both the dot product method and the component-wise method. The final answer is:

Work done 20 units of force * length