Spring's Potential Energy expression

In summary, the conversation discusses deriving an expression for potential energy as a function of compression for a spring attached to a horizontal track. The formula for potential energy is found using the integral of the non-linear force of the spring and is equal to the work done by the spring. This is based on the concept of conservation of energy. The final answer is (1/3 * As^3) + (1/3 * Bs^2).
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Homework Statement


In an experiment, a student wishes to use a spring to accelerate a cart along a horizontal level track. The spring is attached to the left end of the track and produces a non-linear force of magnitude Fs = Ass + Bs, where s is the distance the spring is compressed in meters. A measuring tape, marked in centimeters, is attached to the side of the track.

Derive an expression for the potential energy ∪ as a function of the compression s. Express your answer in terms of A, B, s, and fundamental constants as appropriate.

Homework Equations


Us = 1/2 * k * (Δx)s (changed x to s in this problem)
Fs = -k * Δx (changed x to s in this problem)
ΔE = ∫F * dr

The Attempt at a Solution


So I know that we need to find an expression for potential energy and since they said to included: A, B, s. I wanted a way to include Fs into potential energy. So I set the formula for Fs equal to their version of Fs (so, (-k* Δx) = ( Ass + Bs)) and got k = (Ass + Bs) / Δs. And using this k, I plugged it into Us and got ((Ass + Bs) * Δs) / 2.

However, my teacher showed us that the way you were supposed to do it was using the formula for ΔE using the integral. But why would the ΔE be the same as ∪s? The final answer was (1/3 * As3 ) + (1/3 * Bs2).

Thanks so much for the help! (Additional question about forum, would this count as a intro physics homework or advanced? I posted in advanced but looked at the type of questions and realized that I should've probably posted here. But I want to make sure I place it in the right spot in the future.
Second Note: Saw that the other one was moved, so this is right spot. Sorry for that, promise that it won't happen again.)
 
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  • #2
So you have a non linear force from the spring. Integrating it with respect to r will get you the work it does which in this case will also be equal to the change in potential energy. Think about conservation of energy. You start with potential, subtract the work done by the spring, and end up with less potential. Thus potential equals the work it can do. ##U_i-W=U_f## goes to ##U_f-U_i=-W##
 
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Ohh, wow this make a lot of sense now. Thank You @TJGilb !
 

1. What is spring potential energy and how is it expressed?

Spring potential energy is the energy that a spring possesses when it is compressed or stretched. It is expressed using the formula PE = 1/2kx^2, where k is the spring constant and x is the displacement from equilibrium.

2. How does the spring constant affect the potential energy of a spring?

The spring constant, represented by the letter k, is a measure of how stiff or flexible a spring is. The higher the spring constant, the stiffer the spring and the more potential energy it can store for a given displacement.

3. What is the relationship between the displacement of a spring and its potential energy?

The potential energy of a spring is directly proportional to the square of its displacement from equilibrium. This means that as the displacement increases, the potential energy also increases.

4. Can the potential energy of a spring be negative?

Yes, the potential energy of a spring can be negative. This occurs when the spring is stretched beyond its equilibrium position, resulting in a negative displacement. However, the magnitude of the potential energy is still calculated using the positive value of the displacement.

5. How is the concept of spring potential energy used in real-world applications?

Spring potential energy is used in many real-world applications, such as in springs used in car suspensions, pogo sticks, and trampolines. It is also used in devices such as mousetraps and door hinges. Understanding and manipulating spring potential energy is essential in designing and optimizing these devices for efficient and safe use.

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