Calculating the Work Done by a Force with Vectors

In physics, understanding the work done by a force is fundamental for many practical applications. This article will guide you through the process of calculating the work done by a force given in vector form and the displacement also in vector form, using the example of a force F 2i - j - k and a displacement d 2i - 2j - 5k.

Introduction to Work Done in Physics

The work done by a force is a scalar quantity that represents the energy transfer to or from an object. It can be calculated using the formula:

[W F d]

However, if the force and the displacement are not in the same direction, the work done will be the dot product of the force vector and the displacement vector, given by:

[W F · d Fd cosθ]

where F represents the magnitude of the force, d represents the magnitude of the displacement, and θ is the angle between the force and the displacement vectors.

Understanding the Given Vectors

In the case of the given problem:

F 2i - j - k d 2i - 2j - 5k

Here, i, j, and k represent unit vectors along the x, y, and z axes, respectively.

Calculating the Work Done

To find the work done, we need to determine the dot product of the force vector F and the displacement vector d. The dot product of two vectors is calculated as:

[F · d (2)(2) (-1)(-2) (-1)(-5)]

Carrying out the multiplication and addition:

[F · d 4 2 5 11 units]

However, the original problem statement suggests that the answer is 7 units. This discrepancy may arise from a specific requirement or context that is not considered in the standard formula application.

Components of the Force and Displacement

The work done can also be calculated by finding the component of the force in the direction of the displacement and multiplying it by the magnitude of the displacement. This is represented as:

[fd (2i - j - k) · (2i - 2j - 5k)]

The dot product of the vectors F and d can be expanded as:

[fd (2)(2) (-1)(-2) (-1)(-5) 4 2 5 11 units]

Again, the initial problem states the answer as 7 units, which might be derived from a specific context or simplification not detailed in typical vector operations.

Conclusion

In conclusion, the work done by a force can be calculated using the dot product of the force vector and the displacement vector. The specific case provided in the problem may involve additional simplifications or context that influenced the final value, leading to the answer of 7 units instead of the calculated 11 units.

Additional Resources

If you need further clarification or want to explore more on vector operations and work done in physics, consider the following resources:

Dot Product Calculator Dot Product on Wikipedia Khan Academy - Work Done by a Force