# Harmonic Oscillator 3

1. Jun 3, 2014

### Karol

1. The problem statement, all variables and given/known data
A mass of 0.1 kg has 2 springs of length 20 cm attached to each side, like in the drawing. they are loose. one has a constant of 50 [N/m] and the other 30. The system is between 2 walls 10 cm distant from each spring.
At the second stage the springs are tied each to the nearest wall. the system stabilizes and the springs have different length. what is the length of each spring.
This is the first part of a question, but i got stuck.

2. Relevant equations
Spring force: $$F=kx$$

3. The attempt at a solution
At first i marked with x the distance the k=50 spring stretches. of course it is tied to the wall but i drew it apart in order to clarify. in this notation the k=30 spring stretches 0.1 meter + (0.1-x):
$50x=30\left[ 0.1+\left( 0.1-x \right) \right] \Rightarrow x=0.075$
And it is correct. but then i used an other notation, like in the second drawing, where x is the distance from the wall and tried:
$50(0.1-x)=30(0.1+x)$
And it's wrong, i can't understand why

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2. Jun 3, 2014

### pasmith

$x$ is the distance from the left wall to what? Your diagrams are not exactly clear.

You will get that equation by taking $x$ to be the displacement of the mass from its initial position, with positive $x$ in the direction of the spring with constant 50 N/m.

3. Jun 3, 2014

### tms

It is best to solve problems symbolically, and to only plug in numbers at the very end. That way you can see at each step what is going on.

Try doing that with your second method.

4. Jun 3, 2014

### Karol

Yes, this is another way to look at my second method. your x, the displacement of the mass, is equal to my x, the distance to the wall. of course the mass moves, but to me it was clearer to visualize with the mass fixed.
So, this equation:
$50(0.1-x)=30(0.1+x)$
Describes the mass's displacement x as you said but it's wrong.

5. Jun 3, 2014

### pasmith

Solving that gives $80x = 2$, so $x = 1/40 = 0.025$. The extension in the spring with constant 50 is then $0.1 - x = 0.075$, in agreement with your first method.

6. Jun 3, 2014

### Karol

Now continuing i have to show that the combined spring constant, or as the question says the effective force constant (i hope it's the same as i said) is 80.
But to my understanding it's 30 since one spring works against the other

7. Jun 4, 2014

### pasmith

Work out the equation of motion of the mass, and express it in the form $$m\ddot x = A - Bx.$$ What is $B$?