Calculating Work Done by a Force Vector Along a Displacement Vector

Introduction

In physics, work is a fundamental concept that describes the energy transferred to or from an object via the application of force along a specific displacement. To determine the work done by a force, one can use the formula for work: W mathbf{F} cdot mathbf{d}, where W is the work done, F is the force vector, and d is the displacement vector. The dot product of the two vectors gives the scalar work done.

Understanding the Problem

Given a force vector F 3i 5j N (Newtons) and a displacement vector in the direction of 4i - 3j over a distance of 15 meters, our task is to calculate the work done by this force.

Step-by-Step Methodology

In this section, we will break down the problem into several steps to find the work done by the force vector on the particle.

Step 1: Normalize the Displacement Vector

The first step is to find the unit vector in the direction of the displacement vector d 4i - 3j.

The magnitude of d is given by |mathbf{d}| sqrt{4^2 (-3)^2} sqrt{16 9} sqrt{25} 5

The unit vector (hat{d}) is calculated as: (hat{d} frac{mathbf{d}}{|mathbf{d}|} frac{4mathbf{i} - 3mathbf{j}}{5} frac{4}{5}mathbf{i} - frac{3}{5}mathbf{j})

Step 2: Calculate the Displacement Vector

Next, we calculate the actual displacement vector for a distance of 15 meters in the direction of the unit vector (hat{d}).

The displacement vector is given by (mathbf{d} 15 hat{d} 15 (frac{4}{5}mathbf{i} - frac{3}{5}mathbf{j}) 12mathbf{i} - 9mathbf{j})

Step 3: Calculate the Work Done

Finally, we calculate the work done using the formula for work.

The work done is the dot product of the force vector F and the displacement vector d: (mathbf{F} cdot mathbf{d} (3mathbf{i} 5mathbf{j}) cdot (12mathbf{i} - 9mathbf{j}) 3 cdot 12 5 cdot -9 36 - 45 -9,text{Joules})

Conclusion

The work done by the force is -9 Joules. This negative value indicates that the force is acting in the opposite direction to the displacement, thus doing negative work.

Frequently Asked Questions (FAQs)

1. What is the importance of using the unit vector in calculating work done?

The unit vector is crucial because it gives us the direction of the displacement without the magnitude. This allows us to properly calculate the work even when the force and displacement are in different directions.

2. Can the work done be positive or negative?

Yes, the work done can be positive or negative. A positive work means the force does work in the direction of the displacement, while a negative work indicates the force opposes the direction of the displacement.

3. How do you determine the displacement vector if the force and displacement are given?

First, normalize the direction vector to get the unit vector, then multiply this unit vector by the magnitude of the displacement to find the actual displacement vector.