Calculating Potential Energy of a Spring: A Case Study in Newtonian Dynamics

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The discussion focuses on calculating the potential energy of a spring with a restoring force defined by F{sub x}(x)= -alpha x - beta x^2, where alpha is 60 N/m and beta is 18 N. To find the potential energy function U(x), the relationship U = -∫F dx is suggested, with U set to zero at the equilibrium position. A scenario is presented involving a 0.900 kg mass attached to the spring, which is pulled 1.00 m and released, prompting questions about its speed at 0.500 m from the equilibrium position. Newtonian dynamics is to be applied to determine the object's speed and the instantaneous power at that position. The conversation emphasizes the need for calculus knowledge to solve these problems effectively.
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A certain spring exerts a restoring force F{sub x}(x)= -alpha x - beta x^2 if it is stretched or compressed, where alpha= 60N/m and beta=18N. The mass of the spring is negligible.

How would I : (a) Find the potential energy function U(x) for the spring. Let U=0 when x=0. (b) An object with mass, m = 0.900kg on a frictionless, horizontal surface is attached to this spring, pulled a distance 1.00m to the right to stretch the spring and released. What is the speed of the object when it is 0.500m to the right of the equilibrium position? (c) Use Newtonian dynamics to find the speed at this position. (d) What is the instantaneous power when x= 0.500m?
 
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Hi xshezsciencex! Welcome to PF! :wink:

Hint: potential energy is (minus) work done = -∫ … ? :smile:
 
im even more confused >_<
 
Are you familiar with calculus, or with the equation F=-dU/dx where U is potential energy?
 

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