Calculating work done on nonlinear spring

[kivarocket]

Can anyone help me solve this problem? It seemed straightforward at first, but I am not getting the correct answer of -12 Nm. Thank you!

A nonlinear spring is modeled by a force law given by F(x) = -10x + 3x^2, where F is measured in Newtons and x in meters. How much work is done stretching the spring to x = +2.0 m from its equilibrium position at x = 0.0 m?

2. Attempt: W = F*d; F = -8N, so -8N * 2m = -16J

 

[rock.freak667]

If a force 'F' moves its point of application a distance 'dx', then the incremental work done is dW=F dx, so over an interval x, the entire work done is

∫ dW = ∫ F dx or W = ∫F dx.

 

[kivarocket]

This problem doesn't require calculus, so I don't need to take the integral. However, if I use the equation provided to solve for F, I get -8 N. Since W = F*d (where d = distance traveled), then -8N * 2m = -16 Nm. The correct answer is supposedly -12 Nm, so I'm not sure what I could be missing in this scenario.

 

[Chestermiller]

This problem doesn't require calculus, so I don't need to take the integral. However, if I use the equation provided to solve for F, I get -8 N. Since W = F*d (where d = distance traveled), then -8N * 2m = -16 Nm. The correct answer is supposedly -12 Nm, so I'm not sure what I could be missing in this scenario.

Contrary to what you said, this problem does require calculus. If you follow rock.freak667's advice, you will obtain -12 Nm using calculus.

Chet

 

[rock.freak667]

This problem doesn't require calculus, so I don't need to take the integral. However, if I use the equation provided to solve for F, I get -8 N. Since W = F*d (where d = distance traveled), then -8N * 2m = -16 Nm. The correct answer is supposedly -12 Nm, so I'm not sure what I could be missing in this scenario.

If you draw the graph of F vs x, you will get a parabolic shaped curve. The work done will be the area under that curve from x=0 to x=2 and in order to get the exact area, integral calculus is required otherwise, anything you do will be an approximation and may be inaccurate.

 

[dauto]

This problem doesn't require calculus, so I don't need to take the integral. However, if I use the equation provided to solve for F, I get -8 N. Since W = F*d (where d = distance traveled), then -8N * 2m = -16 Nm. The correct answer is supposedly -12 Nm, so I'm not sure what I could be missing in this scenario.

You're using the formula for the work of a constant force. The force produced by that spring is not constant. You cannot simply use the force obtained at the very end after stretching the spring and use it as if that had been the force produced by the spring all along throughout the whole expansion process. This problem indeed does require calculus.