Find the total work done in moving a particle in the force field given by F = zi +zj + zk along the helix C given by x = cos t ,y = sin t ,z = t from t = 0 to t = pi / 2

To calculate the total work done in moving a particle along a path \( C \) under a force field \( \mathbf{F} \), we can use the line integral of the force field along the curve.

The work done \( W \) is given by:

\[
W = \int_C \mathbf{F} \cdot d\mathbf{r}
\]

Where:
- \( \mathbf{F} = z\mathbf{i} + z\mathbf{j} + z\mathbf{k} \)
- The path \( C \) is parameterized by \( x = \cos t \), \( y = \sin t \), \( z = t \), where \( t \) ranges from 0 to \( \frac{\pi}{2} \).

### Step 1: Parameterize the force field
In terms of \( t \), the force field \( \mathbf{F} \) becomes:
\[
\mathbf{F}(t) = t\mathbf{i} + t\mathbf{j} + t\mathbf{k}
\]

### Step 2: Find the differential of the position vector
The position vector \( \mathbf{r}(t) \) is:
\[
\mathbf{r}(t) = \cos t \mathbf{i} + \sin t \mathbf{j} + t \mathbf{k}
\]

The differential \( d\mathbf{r} \) is the derivative of \( \mathbf{r}(t) \) with respect to \( t \):
\[
d\mathbf{r} = (-\sin t \mathbf{i} + \cos t \mathbf{j} + \mathbf{k}) dt
\]

### Step 3: Calculate the dot product \( \mathbf{F}(t) \cdot d\mathbf{r} \)
Now, compute the dot product of \( \mathbf{F}(t) \) and \( d\mathbf{r} \):

\[
\mathbf{F}(t) \cdot d\mathbf{r} = t(-\sin t) + t(\cos t) + t(1)
\]
\[
\mathbf{F}(t) \cdot d\mathbf{r} = -t \sin t + t \cos t + t
\]

### Step 4: Set up the integral
Now we integrate the dot product from \( t = 0 \) to \( t = \frac{\pi}{2} \):

\[
W = \int_0^{\frac{\pi}{2}} (-t \sin t + t \cos t + t) dt
\]

### Step 5: Compute the integral
We can now calculate the integral term by term:

\[
W = \int_0^{\frac{\pi}{2}} -t \sin t \, dt + \int_0^{\frac{\pi}{2}} t \cos t \, dt + \int_0^{\frac{\pi}{2}} t \, dt
\]

We can solve each of these integrals individually:

1. \( \int t \sin t \, dt \)
2. \( \int t \cos t \, dt \)
3. \( \int t \, dt \)

Let's proceed with these calculations.The total work done in moving the particle along the helix is:

\[
W = -2 + \frac{\pi^2}{8} + \frac{\pi}{2}
\]

If you'd like more details on how the integrals were solved or have any questions, feel free to ask!

Here are 5 related questions for further exploration:
1. How do we calculate work done if the force field were time-dependent?
2. What if the force field had an additional component, say \( F = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \)?
3. How does the nature of the curve \( C \) affect the total work done?
4. How can we visualize the force field and the path together in space?
5. What would be the work done for a different path, such as a straight line from \( t = 0 \) to \( t = \frac{\pi}{2} \)?

**Tip**: For vector fields, the line integral represents the work done along a specific path. Changing the path can significantly alter the result!

Find the total work done in moving a particle in the force field given by F = zi + zj + zk along the helix C given by x = cos t ,y = sin t ,z = t from t = 0 to t = pi / 2.

Calculate the total work done in moving a particle through a vector force field (F = z i + z j + z k) along a helix parameterized by x = cos(t), y = sin(t), and z = t. This problem uses vector calculus concepts such as line integrals and force fields, ideal for undergraduate students studying calculus.

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