CALC An object moving in the xy-plane is subjected to the force F(arrow on top) =(2xy î+x² ĵ) N, where x and y are in m. a. The particle moves from the origin to the point with coordinates (a, b) by moving first along the x -axis to (a, 0) , then parallel to the y -axis. How much work does the force do?

Verified step by step guidance

1. Identify the force components: The force vector given is $\overrightarrow{F}=(2xy\hat{i}+x^2\hat{j})$ N. Here, $\hat{i}$ and $\hat{j}$ are the unit vectors in the x and y directions respectively.
2. Break down the path into two segments: Segment 1 is from (0,0) to (a,0) along the x-axis, and Segment 2 is from (a,0) to (a,b) along the y-axis.
3. Calculate the work done along Segment 1: The work done by a force along a path is given by the integral of the dot product of the force and displacement vectors. For Segment 1, since the displacement is only along the x-axis, the y-component of the force does not contribute to the work. The work done is ${\int }_0^a(2xy)dx$ at y=0.
4. Calculate the work done along Segment 2: For Segment 2, the displacement is only along the y-axis, so only the y-component of the force contributes to the work. The work done is ${\int }_0^b(x^2)dy$ at x=a.
5. Sum the work done over both segments to find the total work done by the force as the object moves from the origin to the point (a, b).

Key Concepts

Work Done by a Force

Work is defined as the integral of force along a path. In this case, the work done by the force F on the object as it moves from one point to another can be calculated by evaluating the line integral of the force vector along the specified path. The work done depends on both the magnitude of the force and the displacement in the direction of the force.

Line Integrals

A line integral is a type of integral that allows us to calculate quantities along a curve. In the context of physics, it is often used to compute work done by a force along a specific path. The line integral of a vector field, such as the force in this problem, requires parameterizing the path and integrating the dot product of the force vector and the differential displacement vector.

Vector Fields

A vector field assigns a vector to every point in a space, representing quantities that have both magnitude and direction. In this problem, the force F is a vector field defined in the xy-plane, where its components depend on the coordinates x and y. Understanding how vector fields behave is crucial for analyzing forces acting on objects and calculating work done along different paths.