Let's solve the problem step by step while addressing all the requested elements.

Given:

1. Path: $y = 0.6x^2$, from $(0, 0)$ to $(2, 2.4)$.
2. Force: $\vec{F} = 3y\hat{i} + 7y\hat{j}$.

We need to calculate the work done along this path.

a. Diagram depicting the trajectory of motion

The path is a parabola defined by $y = 0.6x^2$, and the object travels from $(0, 0)$ to $(2, 2.4)$.

(I will sketch this trajectory later if you need a visual representation.)

b. Equation showing the infinitesimal element of work

The infinitesimal work done $dW$ is given by: $dW = \vec{F} \cdot d\vec{r}$ where $d\vec{r} = dx\hat{i} + dy\hat{j}$ is the infinitesimal displacement vector.

Substitute $\vec{F} = 3y\hat{i} + 7y\hat{j}$ and $d\vec{r}$: $dW = (3y\hat{i} + 7y\hat{j}) \cdot (dx\hat{i} + dy\hat{j})$ $dW = 3y \, dx + 7y \, dy$

c. Mathematical working to find the total work

1. Express $dy$ in terms of $x$:

   Since $y = 0.6x^2$, differentiate to find $dy$: $dy = \frac{d}{dx}(0.6x^2) \, dx = 1.2x \, dx$

2. Substitute $y = 0.6x^2$ and $dy = 1.2x \, dx$ into $dW$:

   $dW = 3y \, dx + 7y \, dy$ $dW = 3(0.6x^2) \, dx + 7(0.6x^2)(1.2x \, dx)$ Simplify: $dW = 1.8x^2 \, dx + 5.04x^3 \, dx$ $dW = (1.8x^2 + 5.04x^3) \, dx$

3. Integrate $dW$ from $x = 0$ to $x = 2$: $W = \int_0^2 (1.8x^2 + 5.04x^3) \, dx$

   Split into two integrals: $W = \int_0^2 1.8x^2 \, dx + \int_0^2 5.04x^3 \, dx$

   Calculate each term: $\int_0^2 1.8x^2 \, dx = 1.8 \cdot \frac{x^3}{3} \Big|_0^2 = 1.8 \cdot \frac{8}{3} = 4.8$ $\int_0^2 5.04x^3 \, dx = 5.04 \cdot \frac{x^4}{4} \Big|_0^2 = 5.04 \cdot \frac{16}{4} = 20.16$

   Add the results: $W = 4.8 + 20.16 = 24.96$

Final Answer:

The total work done is: $\boxed{24.96 \, \text{J}}$

Would you like a visual diagram, detailed explanation, or assistance with any part of this?

Follow-up Questions:

1. How is the concept of work in this problem related to the dot product of vectors?
2. What would happen if the force field $\vec{F}$ varied with $x$ instead of $y$?
3. Why is it important to parametrize the path in terms of $x$?
4. What is the physical meaning of the work done here?
5. Could the work be solved using a numerical method if integration was too complex?

Tip:

When dealing with path integrals, always ensure the force and displacement components align correctly with the curve's parametrization.