Work Done by a Variable Force?

📌 CONCEPT: The work done by a variable force on an object is calculated by integrating the force with respect to the displacement along the direction of the force.

📐 RULE / FORMULA: Work done (W) = ∫[F(x)]dx from x1 to x2, where F(x) is the variable force and x1 and x2 are the initial and final positions of the object.

💡 WORKED EXAMPLE: To find the work done on a 2 kg block moved from x = 0 to x = 4 m by the force F(x) = 3x N, we calculate W = ∫[3x]dx from 0 to 4. Evaluating the integral, we get W = [3/2]x^2 from 0 to 4. This simplifies to W = (3/2)(4^2 - 0^2) = 24 J. The total work done on the block is 24 J, which is converted into the block's kinetic energy at the end of the displacement.

⚠️ COMMON MISTAKE: Students may incorrectly calculate the work done by integrating with respect to time or assuming a constant force, rather than integrating with respect to displacement and considering the variable force.