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

If not, let's just say that the potential energy function for this spring can be found by integrating the restoring force F(x) with respect to x. So U(x)=-∫F(x)dx. Let's call the constant of integration C. Since we want U=0 when x=0, we can set C=0. So the potential energy function for this spring is U(x)=-∫F(x)dx=-∫(-αx-βx²)dx=½αx²+⅓βx³. This is the answer to part (a).For part (b), we know that the total energy (kinetic + potential) of the object-spring system remains constant throughout its
  • #1
xshezsciencex
8
0
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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  • #2
Welcome to PF!

Hi xshezsciencex! Welcome to PF! :wink:

Hint: potential energy is (minus) work done = -∫ … ? :smile:
 
  • #3
im even more confused >_<
 
  • #4
Are you familiar with calculus, or with the equation F=-dU/dx where U is potential energy?
 

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

What is potential energy?

Potential energy is the energy that an object possesses due to its position or state. It is stored energy that can be converted into other forms, such as kinetic energy.

How is potential energy related to a spring?

A spring has potential energy because it is stretched or compressed. When a spring is stretched, it stores potential energy in the form of elastic potential energy. The amount of potential energy stored in a spring depends on the amount it is stretched or compressed.

What is the formula for calculating the potential energy of a spring?

The formula for calculating the potential energy of a spring is PE = 1/2 kx^2, where PE is potential energy, k is the spring constant, and x is the displacement of the spring from its equilibrium position.

What factors affect the potential energy of a spring?

The potential energy of a spring is affected by the spring constant, the displacement of the spring, and the mass of any objects attached to the spring. A higher spring constant or greater displacement will result in a higher potential energy. Adding more mass to the spring will also increase its potential energy.

How is the potential energy of a spring used in everyday life?

The potential energy of a spring is used in many everyday objects, such as trampolines, pogo sticks, and door hinges. It is also used in more complex systems, such as in shock absorbers for vehicles. Understanding the potential energy of a spring is important for designing and using these objects safely and effectively.

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