Stability of a spring mass system in two dimensions

[kyva1929]

Homework Statement

Attached

Homework Equations

5.104 mentioned in the problem is

Also attached

The Attempt at a Solution

Suppose the mass is moved to (x,y), the distance from the mass to the left spring is denoted by A, the distance from the mass to the right spring is denoted by B. I get the expression in the last line.

If I expand the whole expression, and treat sqrt((a+/-x)^2+y^2) as simply a (for small x and y, << a), I can obtain the potential energy in the form of 5.104 after eliminating certain constants, however, that doesn't help me in solving the eqm part of the question.

Differentiating the expression with respect to x doesn't give a solution when the first derivative = 0, which would then indicate that there's no minimum point. I'm stuck. Please help!

 

[hikaru1221]

Then I would like to see how you obtain the simplified expression for U

My result is: $$U = k(a-l_o)^2 + k(1-\frac{l_o}{a})x^2 + k(1-\frac{l_o}{a})y^2$$

Why does $$\partial U/\partial x = 0$$ not give any solution?

 

[kyva1929]

Then I would like to see how you obtain the simplified expression for U

My result is: $$U = k(a-l_o)^2 + k(1-\frac{l_o}{a})x^2 + k(1-\frac{l_o}{a})y^2$$

Why does $$\partial U/\partial x = 0$$ not give any solution?

Can you show me how you arrive at the expression for U? I've tried to derive U from force consideration, the resulting expression looks way too complicated. The expression obtained by merely adding the potential energy from both springs is still too complicated as well.

Thanks for your help :)

 

[hikaru1221]

From your exact expression of U:

$$U = \frac{1}{2}k(\sqrt{a^2+2ax+x^2+y^2}-l_o)^2 + \frac{1}{2}k(\sqrt{a^2-2ax+x^2+y^2}-l_o)^2$$

We have:

$$U = k(a^2+x^2+y^2+l_o^2-2l_o(\sqrt{a^2+2ax+x^2+y^2}+\sqrt{a^2-2ax+x^2+y^2}))$$

Notice that:

$$\sqrt{a^2+2ax+x^2+y^2} \approx a(1+\frac{x}{a}+\frac{x^2}{2a^2}+\frac{y^2}{2a^2})$$

$$\sqrt{a^2-2ax+x^2+y^2} \approx a(1-\frac{x}{a}+\frac{x^2}{2a^2}+\frac{y^2}{2a^2})$$

(due to $$a\sqrt{1+\epsilon} \approx a(1 + \frac{\epsilon}{2})$$ )

And we arrive at the result.

 

[kyva1929]

From your exact expression of U:

$$U = \frac{1}{2}k(\sqrt{a^2+2ax+x^2+y^2}-l_o)^2 + \frac{1}{2}k(\sqrt{a^2-2ax+x^2+y^2}-l_o)^2$$

We have:

$$U = k(a^2+x^2+y^2+l_o^2-2l_o(\sqrt{a^2+2ax+x^2+y^2}+\sqrt{a^2-2ax+x^2+y^2}))$$

Notice that:

$$\sqrt{a^2+2ax+x^2+y^2} \approx a(1+\frac{x}{a}+\frac{x^2}{2a^2}+\frac{y^2}{2a^2})$$

$$\sqrt{a^2-2ax+x^2+y^2} \approx a(1-\frac{x}{a}+\frac{x^2}{2a^2}+\frac{y^2}{2a^2})$$

(due to $$a\sqrt{1+\epsilon} \approx a(1 + \frac{\epsilon}{2})$$ )

And we arrive at the result.

Thank you so much! I'm a newbie and am still not used to approximation technique, thanks for the great help!

 

[Lawrencel2]

From your exact expression of U:

$$U = \frac{1}{2}k(\sqrt{a^2+2ax+x^2+y^2}-l_o)^2 + \frac{1}{2}k(\sqrt{a^2-2ax+x^2+y^2}-l_o)^2$$

We have:

$$U = k(a^2+x^2+y^2+l_o^2-2l_o(\sqrt{a^2+2ax+x^2+y^2}+\sqrt{a^2-2ax+x^2+y^2}))$$

Notice that:

$$\sqrt{a^2+2ax+x^2+y^2} \approx a(1+\frac{x}{a}+\frac{x^2}{2a^2}+\frac{y^2}{2a^2})$$

$$\sqrt{a^2-2ax+x^2+y^2} \approx a(1-\frac{x}{a}+\frac{x^2}{2a^2}+\frac{y^2}{2a^2})$$

(due to $$a\sqrt{1+\epsilon} \approx a(1 + \frac{\epsilon}{2})$$ )

And we arrive at the result.

why did you not treat the constant k as, k_x and k_y??
Can you combine them like that? because i thought that the problem statement given has them as separate constants? help me please

 

[Lawrencel2]

i have this as the exact same question. i was wondering if i could get some clarification on the two different k constants. what is the next step?