Calculating the Work Done by a Force in a 3D Space

In physics, the work done by a force on an object is a crucial concept. This article will guide you through the process of how to calculate the work done by a force in a 3D space using the given formula: W F · d.

Understanding the Formula

The work done by a force F is given by the dot product of the force vector and the displacement vector, d. This can be expressed mathematically as:

W F · d

Where:

W is the work done (in Joules) F is the force vector (in Newtons) d is the displacement vector (in meters)

Step-by-Step Calculation

Let's walk through each step to calculate the work done by a specific force on an object, given its force and displacement vectors in a 3D space.

Step 1: Determine the Displacement Vector

The displacement vector d is calculated from point P (1, 1, 1) to point Q (2, 0, 3) as follows:

d Q - P

Substituting the coordinates:

d (2 - 1) i (0 - 1) j (3 - 1) k

Therefore, the displacement vector is:

d 1 -1 2

Step 2: Write the Force Vector

The force vector is given as:

F 3i - 2j 4k

This can be written in vector form as:

F 3 -2 4

Step 3: Calculate the Dot Product F · d

The dot product of the force vector F and the displacement vector d is calculated as follows:

F · d 3i (-2j) (4k) middot; (1i - 1j 2k)

Breaking it down:

3i middot; 1i 3 -2j middot; -1j 2 4k middot; 2k 8

Adding these together:

F · d 3 2 8 13 J

Conclusion

The work done by the force F is therefore:

W 13 Joules

Frequently Asked Questions (FAQ)

Q1: What is the work done in physics?

A1: Work done in physics is a measure of the energy transferred to or from an object via the application of force along a displacement. The unit of work is the Joule (J).

Q2: How do you determine the displacement vector?

A2: The displacement vector is determined by subtracting the initial position vector from the final position vector. This is done in each dimension (x, y, z).

Q3: What is the dot product?

A3: The dot product of two vectors is a scalar value obtained by multiplying corresponding components of the two vectors and summing the results. It is used to find the angle between two vectors and to calculate work done by a force in a specific direction.

Q4: Why is it important to understand work done?

A4: Understanding the concept of work done is crucial in various fields such as engineering, physics, and robotics. It helps in designing efficient systems, understanding energy transfer, and solving problems related to force and motion.

Q5: Can work done be negative?

A5: Yes, work done can be negative if the force and displacement are in opposite directions. Negative work indicates that the force is acting against the displacement.