Final answer:
The work done by the force as the particle moves from x = 2m to x = 4m is calculated through an integral of the force function and results in 186.67 joules.
Step-by-step explanation:
The question deals with the calculation of the work done by a variable force when a particle moves from one position to another. Since force is given as a function of position (F = 10x² N), the work done can be calculated by integrating the force over the path from the initial to the final position. The integral of F = 10x² from x = 2m to x = 4m will give us the work done.
To find the work done, we use the formula W = ∫ F dx, where W is the work done, F is the force as a function of x, and dx is the differential element of the path in the direction of the force. The integration of 10x² from 2 to 4 is:
∫2²4 10x² dx = [10x³/3]2²4 = (10/3)(4³ - 2³) = (10/3)(64 - 8) = (10/3)(56) = 186.67 J
Hence, the work done by the force as the particle moves from x = 2m to x = 4m is 186.67 joules.