Question

The force on a particle is described by 2x³-1 pounds at a point x feet along the x-axis. Find the work done in moving the particle from the origin to x=2?

Answer 1

The work done to move a particle along the x-axis from the origin to x=2 under the force described by F(x) = 2x^3 - 1 pounds is 6 pounds-feet, calculated by integrating the force over the distance travelled.

Step-by-step explanation:

The student has asked about the work done by a force on a particle as it moves along the x-axis. Work is defined as the integral of the force with respect to displacement. In this case, the force acting on the particle varies with its position as described by the force function F(x) = 2x3 - 1 pounds, and the particle moves from the origin (x=0) to x=2 feet. The work W is given by the definite integral of F(x) with respect to x from 0 to 2.

The calculation of work involves integrating the force function:

1. Set up the integral: W = ∫ F(x) dx from 0 to 2.
2. Calculate the integral: W = ∫ (2x3 - 1) dx from 0 to 2.
3. Integrate and evaluate from 0 to 2: W = [0.5x4 - x]20.
4. Find the result: W = (0.5(2)4 - 2) - (0.5(0)4 - 0) = 6 pounds-feet.

The work done to move the particle from the origin to x=2 is thus 6 pounds-feet.