Work Done By Force Field On A Helix Path A Step-by-Step Guide

Hey everyone! Today, we're diving into a fun problem involving force fields and particle motion. We're going to calculate the work done by a specific force field on a particle as it moves along a helix. This is a classic example in physics and calculus that combines vector calculus with parametric equations, so buckle up and let's get started!

Problem Statement

We're given a force field defined as $F(x, y, z) = 6xi + 6yj + 3k$. Think of this as a field of forces acting at every point in space. Imagine you have a particle moving through this field, and the field is either pushing or pulling on it. We also have a particle moving along a helix described by the parametric equation $r(t) = 7 ext{cos}(t)i + 7 ext{sin}(t)j + 2tk$, where $0 ext{≤} t ext{≤} 2π$. This helix essentially spirals around the z-axis as t increases. Our mission, should we choose to accept it, is to find the total work done by the force field on the particle as it traverses this helical path.

Understanding the Concepts

Before we jump into the calculations, let's make sure we're all on the same page with the underlying concepts. The key here is the relationship between work, force, and displacement in a vector field.

Work Done by a Force Field

In physics, work is done when a force causes a displacement. When the force is constant and the displacement is along a straight line, work is simply the dot product of the force vector and the displacement vector. However, when the force varies or the path is curved (like our helix!), we need to use an integral to sum up the tiny bits of work done along the path. Mathematically, the work done by a force field $F$ along a curve $C$ is given by the line integral:

$W = ∫_C F ⋅ dr$

Where:

• $W$ is the work done.
• $F$ is the force field (a vector field).
• $dr$ is an infinitesimal displacement vector along the curve $C$.
• The integral is taken along the curve $C$.

Parametric Representation of a Curve

Our curve, the helix, is given in parametric form as $r(t) = 7 ext{cos}(t)i + 7 ext{sin}(t)j + 2tk$. This means that the position of the particle at any time $t$ is given by the vector $r(t)$. The parameter $t$ varies from $0$ to $2π$, tracing out one full turn of the helix. To evaluate the line integral, we need to express everything in terms of the parameter $t$.

Finding $dr$

The infinitesimal displacement vector $dr$ represents a tiny step along the curve. In parametric form, we can find $dr$ by taking the derivative of the position vector $r(t)$ with respect to $t$ and multiplying by $dt$:

$dr = r'(t) dt$

This is a crucial step because it allows us to convert the line integral along the curve into a regular integral with respect to the parameter $t$.

Setting Up the Integral

Okay, now we've got the concepts down, let's set up the integral to calculate the work done. This involves several steps, but don't worry, we'll take it nice and slow.

Step 1: Find $r'(t)$

First, we need to find the derivative of the position vector $r(t)$ with respect to $t$. This will give us the tangent vector to the curve at any point, which is essential for calculating $dr$.

$r(t) = 7 ext{cos}(t)i + 7 ext{sin}(t)j + 2tk$

Taking the derivative with respect to $t$, we get:

$r'(t) = -7 ext{sin}(t)i + 7 ext{cos}(t)j + 2k$

Step 2: Express $F$ in terms of $t$

Next, we need to express the force field $F$ in terms of the parameter $t$. Remember, $F$ is given as a function of $x$, $y$, and $z$, but our curve is described by $r(t)$, which gives $x$, $y$, and $z$ in terms of $t$.

From $r(t)$, we have:

• $x = 7 ext{cos}(t)$
• $y = 7 ext{sin}(t)$
• $z = 2t$

Substitute these into the force field equation:

$F(x, y, z) = 6xi + 6yj + 3k$

$F(t) = 6(7 ext{cos}(t))i + 6(7 ext{sin}(t))j + 3k$

$F(t) = 42 ext{cos}(t)i + 42 ext{sin}(t)j + 3k$

Now our force field is expressed as a function of $t$.

Step 3: Calculate $F ⋅ r'(t)$

Now comes the crucial step: calculating the dot product of the force field $F(t)$ and the derivative of the position vector $r'(t)$. This dot product will give us a scalar function of $t$, which we can then integrate.

$F(t) ⋅ r'(t) = (42 ext{cos}(t)i + 42 ext{sin}(t)j + 3k) ⋅ (-7 ext{sin}(t)i + 7 ext{cos}(t)j + 2k)$

Remember, the dot product of two vectors is calculated as the sum of the products of their corresponding components:

$F(t) ⋅ r'(t) = (42 ext{cos}(t))(-7 ext{sin}(t)) + (42 ext{sin}(t))(7 ext{cos}(t)) + (3)(2)$

$F(t) ⋅ r'(t) = -294 ext{cos}(t) ext{sin}(t) + 294 ext{sin}(t) ext{cos}(t) + 6$

Notice that the first two terms cancel each other out:

$F(t) ⋅ r'(t) = 6$

Wow, that simplified nicely! The dot product is simply a constant value of 6.

Step 4: Set Up the Definite Integral

We're almost there! Now we can set up the definite integral to calculate the work done. Recall the formula for work:

$W = ∫_C F ⋅ dr$

We've found that $F(t) ⋅ r'(t) = 6$, and we know that $dr = r'(t) dt$. Also, our parameter $t$ ranges from $0$ to $2π$. So, our integral becomes:

$W = ∫_0^{2π} 6 dt$

Evaluating the Integral

Okay, the integral is set up, and now we just need to evaluate it. This is a straightforward integral:

$W = ∫_0^{2π} 6 dt$

$W = 6 ∫_0^{2π} dt$

$W = 6 [t]_0^{2π}$

$W = 6 (2π - 0)$

$W = 12π$

The Result

And there we have it! The work done by the force field $F(x, y, z) = 6xi + 6yj + 3k$ on the particle as it moves along the helix $r(t) = 7 ext{cos}(t)i + 7 ext{sin}(t)j + 2tk$ from $0 ext{≤} t ext{≤} 2π$ is:

$W = 12π$ units of work.

So, the final answer is 12π. We successfully navigated through the force field, calculated the infinitesimal work at each point along the helical path, and summed it all up using integration. This problem beautifully illustrates the power of vector calculus in solving real-world physics problems.

Key Takeaways

Let's recap the key steps we took to solve this problem:

1. Understanding the Concept of Work: We started by understanding that work done by a force field along a curve is given by the line integral $W = ∫_C F ⋅ dr$.
2. Parametric Representation: We used the parametric equation of the helix, $r(t)$, to describe the particle's position as a function of the parameter $t$.
3. Finding $dr$: We calculated $dr$ by taking the derivative of $r(t)$ with respect to $t$ and multiplying by $dt$ ($dr = r'(t) dt$).
4. Expressing $F$ in Terms of $t$: We substituted the parametric equations for $x$, $y$, and $z$ into the force field equation to express $F$ as a function of $t$.
5. Calculating the Dot Product: We calculated the dot product $F(t) ⋅ r'(t)$, which gave us a scalar function of $t$.
6. Setting Up and Evaluating the Integral: We set up the definite integral $W = ∫_0^{2π} F(t) ⋅ r'(t) dt$ and evaluated it to find the work done.

By following these steps, we were able to solve a seemingly complex problem by breaking it down into smaller, manageable parts. Remember, practice makes perfect, so keep working on these types of problems, and you'll become a pro in no time!

If you have any questions or want to explore similar problems, feel free to ask. Keep exploring the fascinating world of physics and calculus, guys!