The Force Field and the Helix

Oct 13, 2025 by windailys.com 25 views

Introduction

In physics, the work done by a force field on an object is a fundamental concept that helps us understand the energy transferred between the object and the force field. When a particle moves through a force field, the force field exerts a force on the particle, causing it to accelerate. The work done by the force field is a measure of the energy transferred to the particle as it moves through the force field. In this article, we will find the work done by the force field $F(x, y, z) = 5x \, \mathbf{i} + 5y \, \mathbf{j} + 4 \, \mathbf{k}$ on a particle that moves along the helix $r(t) = 4 \cos(t) \, \mathbf{i} + 4 \sin(t) \, \mathbf{j} + 7t \, \mathbf{k}, \, 0 \leq t \leq 2\pi$ .

The Force Field and the Helix

The force field $F(x, y, z) = 5x \, \mathbf{i} + 5y \, \mathbf{j} + 4 \, \mathbf{k}$ is a vector field that depends on the position of the particle in space. The helix $r(t) = 4 \cos(t) \, \mathbf{i} + 4 \sin(t) \, \mathbf{j} + 7t \, \mathbf{k}, \, 0 \leq t \leq 2\pi$ is a curve in space that the particle follows as it moves. To find the work done by the force field on the particle, we need to calculate the dot product of the force field and the velocity vector of the particle at each point on the helix.

Calculating the Velocity Vector

The velocity vector of the particle is the derivative of the position vector with respect to time. In this case, the position vector is $r(t) = 4 \cos(t) \, \mathbf{i} + 4 \sin(t) \, \mathbf{j} + 7t \, \mathbf{k}$ . Taking the derivative of this vector with respect to time, we get:

\frac{dr}{dt} = -4 \sin(t) \, \mathbf{i} + 4 \cos(t) \, \mathbf{j} + 7 \, \mathbf{k}

This is the velocity vector of the particle at each point on the helix.

Calculating the Dot Product

The dot product of the force field and the velocity vector is given by:

F(r(t)) \cdot \frac{dr}{dt} = (5x \, \mathbf{i} + 5y \, \mathbf{j} + 4 \, \mathbf{k}) \cdot (-4 \sin(t) \, \mathbf{i} + 4 \cos(t) \, \mathbf{j} + 7 \, \mathbf{k})

Substituting the expressions for $x$ , $y$ , and $z$ in terms of $t$ , we get:

F(r(t)) \cdot \frac{dr}{dt} = (5(4 \cos(t)) \, \mathbf{i} + 5(4 \sin(t)) \, \mathbf{j} + 4 \, \mathbf{k}) \cdot (-4 \sin(t) \, \mathbf{i} + 4 \cos(t) \, \mathbf{j} + 7 \, \mathbf{k})

Expanding the dot product, we get:

F(r(t)) \cdot \frac{dr}{dt} = -20 \sin(t) \cos(t) + 20 \sin(t) \cos(t) + 28

Simplifying the expression, we get:

F(r(t)) \cdot \frac{dr}{dt} = 28

Calculating the Work Done

The work done by the force field on the particle is given by the integral of the dot product of the force field and the velocity vector with respect to time:

W = \int_{0}^{2\pi} F(r(t)) \cdot \frac{dr}{dt} dt

Substituting the expression for the dot product, we get:

W = \int_{0}^{2\pi} 28 dt

Evaluating the integral, we get:

W = 28(2\pi - 0)

Simplifying the expression, we get:

W = 56\pi

Therefore, the work done by the force field on the particle that moves along the helix is $56\pi$ .

Conclusion

$In this article, we found the work done by the force field $F(x, y, z) = 5x \, \mathbf{i} + 5y \, \mathbf{j} + 4 \, \mathbf{k}$ on a particle that moves along the helix $r(t) = 4 \cos(t) \, \mathbf{i} + 4 \sin(t) \, \mathbf{j} + 7t \, \mathbf{k}, \, 0 \leq t \leq 2\pi$. The work done is given by the integral of the dot product of the force field and the velocity vector with respect to time. We found that the work done is $56\pi$. This result can be used to understand the energy transferred between the particle and the force field as the particle moves along the helix.<br/> **Work Done by a Force Field on a Particle Moving Along a Helix: Q&A**$

Introduction

In our previous article, we found the work done by the force field $F(x, y, z) = 5x \, \mathbf{i} + 5y \, \mathbf{j} + 4 \, \mathbf{k}$ on a particle that moves along the helix $r(t) = 4 \cos(t) \, \mathbf{i} + 4 \sin(t) \, \mathbf{j} + 7t \, \mathbf{k}, \, 0 \leq t \leq 2\pi$ . In this article, we will answer some common questions related to the work done by a force field on a particle moving along a helix.

Q: What is the work done by a force field on a particle?

A: The work done by a force field on a particle is a measure of the energy transferred between the particle and the force field as the particle moves through the force field. It is calculated by integrating the dot product of the force field and the velocity vector of the particle with respect to time.

Q: How do you calculate the work done by a force field on a particle?

A: To calculate the work done by a force field on a particle, you need to follow these steps:

Find the force field and the velocity vector of the particle.
Calculate the dot product of the force field and the velocity vector.
Integrate the dot product with respect to time to find the work done.

Q: What is the significance of the work done by a force field on a particle?

A: The work done by a force field on a particle is significant because it helps us understand the energy transferred between the particle and the force field. It can also be used to calculate the force required to move a particle along a certain path.

Q: Can the work done by a force field on a particle be negative?

A: Yes, the work done by a force field on a particle can be negative. This occurs when the force field and the velocity vector of the particle are in opposite directions.

Q: How does the work done by a force field on a particle depend on the force field and the velocity vector?

A: The work done by a force field on a particle depends on both the force field and the velocity vector of the particle. The force field determines the direction and magnitude of the force exerted on the particle, while the velocity vector determines the direction and magnitude of the particle's motion.

Q: Can the work done by a force field on a particle be zero?

A: Yes, the work done by a force field on a particle can be zero. This occurs when the force field and the velocity vector of the particle are perpendicular to each other.

Q: How does the work done by a force field on a particle relate to the energy of the particle?

A: The work done by a force field on a particle is related to the energy of the particle. The work done can increase or decrease the energy of the particle, depending on the direction of the force field and the velocity vector.

Conclusion

In this article, we answered some common questions related to the work done by a force field on a particle moving along a helix. We hope that this article has provided a better understanding of the concept of work done by a force field on a particle and its significance in physics.

Additional Resources

For more information on the work done by a force field on a particle, we recommend the following resources:

[1] "Work and Energy" by HyperPhysics
[2] "Work Done by a Force Field" by Physics Classroom
[3] "Work and Energy" by Khan Academy

We hope that these resources will provide a more in-depth understanding of the concept of work done by a force field on a particle.