# Finding the extension of a spring

1. Oct 8, 2014

### Dustinsfl

1. The problem statement, all variables and given/known data
A weight of $50$ N is suspended from a spring of stiffness $4000$ N/m and is subjected to a harmonic force of amplitude $60$ N and frequency $6$ Hz.

2. Relevant equations

3. The attempt at a solution
Since $W = mg = 50$, we have that the suspended mass, $m = 5.10204$, and we know that $f = \frac{\omega}{2\pi} = 6$ so $\omega = 12\pi$. The harmonic forcing term is then
$$F(t) = 60\cos(12\pi t)$$
and our equation of motion is
$$\ddot{x} + \frac{4000}{5.10204}x = \frac{60}{5.10204}\cos(12\pi t).$$
Solving the transient and steady solution, we obtain
$$x(t) = A\cos(28t) + B\sin(28t) - 0.0184551\cos(12\pi t)$$
How do I determine the extension of spring from the suspended mass? This value would then be $x(0) = x_0$. Additionally, I will assume any motion starts from rest so $\dot{x}(0) = 0$ which leads to $B = 0$ and $A$ can be defined as $x_0 - \frac{F_0}{k - m\omega^2}$ where $\omega = 12\pi$ and $F_0 = 60$
$$x(t) = (x_0 + 0.0184551)\cos(28t) - 0.0184551\cos(12\pi t)$$
Would the extension of the spring simply be, $F = kx$ where $F = 50$ so
$$x_0 = \frac{F}{k} = \frac{1}{80}\mbox{?}$$

2. Oct 8, 2014

### Orodruin

Staff Emeritus
1/80 what? Lightyears? Attometers? Quoting units is very important in physics ...

You problem statement is missing a question so it is difficult to understand exactly what the problem you wish to solve is.

For the equilibrium point, the forces of gravity and from the spring need to cancel. Which leads to the equation yoy are quoting in the end.

3. Oct 8, 2014

### nasu

What is the question here? What are you trying to do?

4. Oct 8, 2014

### Dustinsfl

What is the extension of the spring? The units are trivial to solve for N/(N/m).

5. Oct 8, 2014

### Dustinsfl

What is the extension of the spring?

6. Oct 8, 2014

### Orodruin

Staff Emeritus
The point is that units are important. I am perfectly aware of how they propagate but you need to provide your values with units throughout. Your answer is not 1/80, it is 1/80 m. Just saying 1/80 has no meaning as a length.

Extension of the spring when? Based on the information you have given it is oscillating. Do you need to find it as a function of time or the average extension? This is why we ask you to provide the problem exactly as stated word by word.

7. Oct 8, 2014

### Dustinsfl

I know it is oscillating. The question just says "the extension of the spring due to the suspended weight"

8. Oct 8, 2014

### Staff: Mentor

What Orodruin is getting at is: where is the mass located to start with (at t = 0), what is the velocity of the mass initially (at t = 0), and what is the force as a function of time (starting from t = 0)?

Chet

9. Oct 8, 2014

### Dustinsfl

$x(0) = x_0$ not giving, $\dot{x}(0) = \dot{x}_0$ not giving but I am assuming it is zero, and the forcing function $F(t)$ is in the statement. I believe finding the springs extension is going to be the initial position.

10. Oct 8, 2014

### Staff: Mentor

For the initial conditions you assumed, it looks like you did the problem correctly. So what is it you want us to help you with?

Chet

11. Oct 8, 2014

### Dustinsfl

I just wanted to know how to find the extension of the spring due to the suspended mass. I think it is just $x = \frac{50}{4000}$ so I was checking whether my though is correct.

12. Oct 8, 2014

### Staff: Mentor

With no additional information, that's what I would have used for the initial extension.

Chet

13. Oct 8, 2014

### nasu

If this is the only question, then all the formulas you wrote there, with the exception of last one, are irrelevant. You must admit that your post was a little confusing.