Power of a Variable Force?

📌 CONCEPT: The work done by a variable force on an object is given by the definite integral of the force function with respect to the displacement of the object.

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

💡 WORKED EXAMPLE: For the given problem, F(t) = 2t N, we need to find the work done in moving the particle from x = 1 m to x = 3 m. To do this, we need to convert the force function from a function of time (t) to a function of position (x), which is F(x) = 2x. Then, we can evaluate the integral W = ∫[1, 3] 2x dx, which gives W = [x^2] from 1 to 3 = 9 - 1 = 8 J.

⚠️ COMMON MISTAKE: Students often forget to convert the force function from a function of time to a function of position before evaluating the integral, or they may incorrectly evaluate the integral.