Bead attached to a spring and moving along a horizontal wire

[beowulf.geata]

I'm self-studying an introductory book on mathematical methods and models and came across the following problem:

1. A bead of mass m is threaded onto a frictionless horizontal wire. The bead is attached to a model spring of stiffness k and natural length l₀, whose other end is fixed to a point A at a vertical distance h from the wire (where h > l₀). The position x of the bead is measured from the point on the wire closest to A. Find the potential energy function U(x).

Homework Equations

I'm rather puzzled by the solution given in the book, which claims that since the length of the spring is (h²+x²)^1/2 and its extension is (h²+x²)^1/2 - l₀, then U(x) = (1/2)k((h²+x²)^1/2 - l₀)².

The Attempt at a Solution

I think that's incorrect because only the x-component of the force exerted by the spring on the bead is relevant to the calculation of U(x). The x-component is -k(l-l₀)cos$\theta$, where l = (h²+x²)^1/2 and cos$\theta$ = x/(h²+x²)^1/2. Hence, U(x) is -$\int$(-kx + kl₀x/(h²+x²)^1/2)dx. Is this correct?

 

[CWatters]

Humm. Interesting question. I'm not sure I can answer it. At first thought you are correct, but then ...

With the spring stretched the energy stored in the spring will be that calculated in the book. What happens to the energy when the spring is released? Where does it go if you apply conservation of energy? The wire is frictionless so as far as I can see all the energy ends up in the bead (eg not some fraction of that due to the Cosθ issue you raise).

So I conclude the book is correct but I'm willing to be convinced you are right and the book is wrong.

 

[TSny]

See if you can show that both expressions are correct, although they might differ by an additive constant.

 

[beowulf.geata]

The integral should evaluate to

kx²/2 - kl₀(h² + x²)^1/2 + C.

Hence, the difference between my solution and the book's is:

kx²/2 - kl₀(h² + x²)^1/2 + C - (k(h² + x²)^1/2l₀ + (1/2)kh² + (1/2)kx² - (1/2)kl₀²)
= C - (1/2)kh² + (1/2)kl₀²,

so the two solutions do appear to differ by a constant...

 

[Nickie]

I would like to clarify that both solutions presented in the book and your attempt at a solution are correct, but they are solving for different potentials. The solution in the book is finding the potential energy function U(x) for the entire system, including the potential energy stored in the spring and the gravitational potential energy due to the vertical distance h. This is why the length of the spring and its extension are taken into account in the calculation.

On the other hand, your solution is finding the potential energy function U(x) solely for the spring, without considering the gravitational potential energy. This is a valid approach, but it will result in a different potential energy function.

Both solutions are correct depending on the context and what potential energy is being considered. As a scientist, it is important to understand the assumptions and context of a problem in order to choose the appropriate approach and solution.