To calculate the total work done in moving a particle through a force field along a given path, we can use the line integral of the force field along the path. The work $W$ is given by:

$W = \int_C \mathbf{F} \cdot d\mathbf{r}$

Where:

• $\mathbf{F} = (F_x, F_y, F_z) = 3x \cdot y \hat{i} - 52 \hat{j} + 10x \hat{k}$ is the force field.
• $\mathbf{r}(t) = (x(t), y(t), z(t))$ is the parameterized path along which the particle moves.
• $d\mathbf{r} = \frac{d\mathbf{r}}{dt} dt$ is the differential displacement along the path.

Given: $x(t) = 1 + t^2, \quad y(t) = 24^2, \quad z(t) = f^3$

First, let's compute the derivatives $\frac{dx}{dt}$, $\frac{dy}{dt}$, and $\frac{dz}{dt}$:

$\frac{dx}{dt} = 2t, \quad \frac{dy}{dt} = 0 \quad (\text{since } y(t) = \text{constant}), \quad \frac{dz}{dt} = 3f^2 \cdot \frac{df}{dt}$

Now, the differential displacement $d\mathbf{r}$ is:

$d\mathbf{r} = \left(2t \hat{i}, 0 \hat{j}, 3f^2 \frac{df}{dt} \hat{k} \right) dt$

Now, compute $\mathbf{F} \cdot d\mathbf{r}$:

$\mathbf{F} \cdot d\mathbf{r} = \left(3x \cdot y \right) \cdot 2t \, dt + (-52) \cdot 0 \, dt + 10x \cdot 3f^2 \frac{df}{dt} \, dt$

Substitute $x(t) = 1 + t^2$ and $y = 576$ (since $y = 24^2$):

$\mathbf{F} \cdot d\mathbf{r} = \left(3(1+t^2) \cdot 576 \right) \cdot 2t \, dt + 10(1 + t^2) \cdot 3f^2 \frac{df}{dt} \, dt$

Simplify the integrand:

$\mathbf{F} \cdot d\mathbf{r} = \left(3456t(1+t^2) + 30f^2(1 + t^2) \frac{df}{dt} \right) dt$

Now, integrate from the initial point to the final point. The limits for $t$ and $f$ are given by the conditions $1/x = 1$ and $f = 2$. Therefore, at the initial point:

• $t_1 = 0$ (since $x(0) = 1 + 0^2 = 1$)
• $t_2$ is not given, but can be solved if $x = 1 + t^2$, which depends on the value of $t_2$ when $f = 2$.

Therefore, we can calculate the work $W$ by integrating:

$W = \int_{t_1}^{t_2} \left(3456t(1+t^2) + 30f^2(1 + t^2) \frac{df}{dt} \right) dt$

Without specific bounds on $t$, a general solution can be complex. The above setup, however, is how you would compute it if given specific bounds.

Would you like to explore more details or discuss any specific part of this? Here are some questions to consider:

1. What is the exact value of $f(t)$ to find $t_2$ from the given path equations?
2. How do you solve for $t_2$ when the relationship between $f$ and $t$ is given?
3. Can we assume specific limits for $t_1$ and $t_2$ based on $f(0) = 0$ and $f = 2$?
4. How do you approach this integral when $f(t)$ is explicitly given?
5. Would you like a numerical approximation if explicit values for limits are provided?

Tip: Always check the parametrization bounds carefully when dealing with line integrals to ensure all variables are correctly accounted for.