1. ## Equation of motion

Need help finding the equation of motion. Have no idea where to even start.

2. ## Re: Equation of motion

If you have no idea, why were you given the problem?

3. ## Re: Equation of motion

I can see the attachment but it doesn't help clarify your question. The entries in the attachment look correct, at least for part (c). So, what exactly is your question?

4. ## Re: Equation of motion

I do not understand how to find the equation of motion

5. ## Re: Equation of motion

Originally Posted by shunae95
I do not understand how to find the equation of motion
So type out exactly what you are asking. Why do you think that we should struggle to read your postings?

6. ## Re: Equation of motion

A mass of 200 grams stretches a spring 49/80 meters.
I've found the spring constant k.
m= 200g = .2kg
L= 49/80 = .6125

k= 3.2

w^2= 16

I've found w, T, and f for the free undamped motion.
w= 4
T= 1.57
f= .637

The general solution, I've found to be
u(t) = c1cos(4t)+c2sin(4t)

So now the question is: Suppose that the mass initially starts at rest in its equilibrium position. Solve for the equation of motion.

7. ## Re: Equation of motion

So far so good. At time t=0 the spring is in the equilibrium position, so u(0) = 0. From that it must be that the unknown constant c1 = 0, and your equation is now:

$u(t) = C_2 \sin(\omega t)$.

Now take the derivative:

$u'(t) =\omega C_2 \cos(\omega t)$

At time t = 0 you say that the mass is at rest, so u'(0) = 0. Which means c2 = 0.

So the equation of motion is: u(t) = 0.

In other words it doesn't move at all. If a spring/mass is in the equilibrium position and is not moving, there's nothing that's going to make it start moving. Hence u(t) = 0. Usually for these type of problems they give you an initial deflection, or velocity, or acceleration, but in this case it's simply at rest.

8. ## Re: Equation of motion

Thanks for you help. I tried working it out and trying your solution but that still did not give me the correct answer.

9. ## Re: Equation of motion

Originally Posted by shunae95
Thanks for you help. I tried working it out and trying your solution but that still did not give me the correct answer.
Please write the problem out exactly as stated. As I noted the problem as you wrote it is a bit strange - I've never seen a problem like this where there is no initial displacement and no initial velocity. But I wonder - the problem says the mass starts at its "equilibrium position," but perhaps what they meant to say is that it starts at its "unstretched position?"

10. ## Re: Equation of motion

OK, I see what the problem is. Looking at the attachment it appears that this problem is a continuation of part (c), not a problem on its own. So there is an exterior force of 6 sin(wt) acting on the mass. The basic equation governing all this is:

$\sum F = ma$

so,

$6 sin(2 t) - kx = m\ddot x$

Rearrange:

$\frac {6 \sin(2t)} m = \ddot x + \frac k m x$

Using the coefficients you already worked out yields:

$\frac {6 \sin(2t)}{0.2 kg} = 12A \cos(2t) + 12B \sin(2 t)$

from which

$A = 0, \ B=\frac {30}{12} = 2.5$

You have been told that the equation of motion is $x(t) = A \cos(2t) + B \sin(2t)$. Can you take it from here?