Greetings, How do I properly intergrate the follwing? A conservative force F = (

Question

Greetings, How do I properly intergrate the follwing? A conservative force F = ((7.0 N/m)x - 8.0N)i acts on aparticle that moves on an x axis. The PE associated with this forceis assigned a value of 26 J at x=0. What is the max PE? Where onthe axis is the PE = 0? Thanks! Greetings, How do I properly intergrate the follwing? A conservative force F = ((7.0 N/m)x - 8.0N)i acts on aparticle that moves on an x axis. The PE associated with this forceis assigned a value of 26 J at x=0. What is the max PE? Where onthe axis is the PE = 0? Thanks!

Explanation / Answer

Hi, given force is F = ((7.0 N/m)x - 8.0N)i But potential energy is the negative of line integral of force then integrating the force we get U = - Integral [ ((7.0 N/m)x -8.0N) dx]                                                     = -[(3.5N/m2)x2 - (8N/m)x] +C Given at x =0 U = 26J therefore C = 26J Then U = (8N/m)x - (3.5N/m2)x2 + 26J Now for U to be maximum dU/dx = 0 Then ((7.0 N/m)x - 8.0N) = 0 or x = 8/7m = 1.143m Let U = 0 Then (8N/m)x - (3.5N/m2)x2 + 26J = 0 solving for x we get x= -1.81m and x = 4.09m Hope this helps you.

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