Differential Equations dealing with spring physics

[SauerKrauter]

I am greatly struggling with a homework assignment given out by my physics professor. It mostly differential equations but based on spring physics. I'll type out the first couple parts but will most likely need help with more as I get farther.

Homework Statement

Y(t) : the y position of mass M on spring.
L₀ : initial position of mass M.
K : the spring constant of spring.
B : the coefficient of friction.

show that Y(t) observes the differential equation Y''(t)+(b/m)y'(t)+(k/m)Y(t) = 0 and
find the general solution using Y(t) = const * e^ikt with k related to m, k, and b.

Homework Equations

F_spring = mY''(t) = -kY(t)
F_fric = -BY'(t)

The Attempt at a Solution

I'm thinking that the differential equation holds true if and only if the force of friction (and therefore Y'(t)) is equal to zero, because Y''(t) and (k/m)Y(t) are equal and opposite and therefore must add up to zero.

For solving the general solution of the differential equation I can get it the form Ce^ikt[i²a²+(b/m)ia+(k/m)]=0 but am not sure where to go from there. Clearly its this part of the question that's more difficult and I would love if someone could help me out with it, much more so than the above.

 

[CAF123]

To show the eqn given, I think you have to just draw a free body diagram for the mass attached to the spring, identifying all the forces. Then write NII and the result just comes out.

 

[SauerKrauter]

The homework really has nothing to do with solving for forces, just solving for the general solution of the differential equations, so free body diagrams really don't do anything for me here.

 

[rude man]

I am greatly struggling with a homework assignment given out by my physics professor. It mostly differential equations but based on spring physics. I'll type out the first couple parts but will most likely need help with more as I get farther.

Homework Statement

Y(t) : the y position of mass M on spring.
L₀ : initial position of mass M.
K : the spring constant of spring.
B : the coefficient of friction.

show that Y(t) observes the differential equation Y''(t)+(b/m)y'(t)+(k/m)Y(t) = 0 and
find the general solution using Y(t) = const * e^ikt with k related to m, k, and b.

Homework Equations

F_spring = mY''(t) = -kY(t)
F_fric = -BY'(t)

The Attempt at a Solution

I'm thinking that the differential equation holds true if and only if the force of friction (and therefore Y'(t)) is equal to zero, because Y''(t) and (k/m)Y(t) are equal and opposite and therefore must add up to zero.

Not so. You must consider all forces acting on the mass. That includes the spring AND the damper. So you need to add the BY' term to your eq.

Then trust in your given solution form and solve.

 

[haruspex]

B : the coefficient of friction.
show that Y(t) observes the differential equation Y''(t)+(b/m)y'(t)+(k/m)Y(t) = 0 and

There might be something in the set-up that needs more explanation. Where is this friction coming from? Usually it's taken to be proportional to the normal force, not depending on the speed. If you're not told to assume it's proportional to speed, I don't see how you can arrive at that equation.

 

[rude man]

There might be something in the set-up that needs more explanation. Where is this friction coming from? Usually it's taken to be proportional to the normal force, not depending on the speed. If you're not told to assume it's proportional to speed, I don't see how you can arrive at that equation.

Good point. This is not a damper, it's friction. Static AND dynamic! A problem for the simulator, not the pen! And definitely NOT modelable by the ODE given the OP.

 

[SauerKrauter]

I'm told to assume its proportional to speed, the equations i gave are written right on the homework and are the only things written on the homework, along with a small picture. I promise you anything that seems off was not something I did but something weird in the assignment from my professor.

 

[CAF123]

I'm told to assume its proportional to speed, the equations i gave are written right on the homework and are the only things written on the homework, along with a small picture. I promise you anything that seems off was not something I did but something weird in the assignment from my professor.

Is the set up vertical?

 

[SauerKrauter]

Horizontal, with the mass not looking like it touches the ground in the picture.

 

[rude man]

I'm told to assume its proportional to speed, the equations i gave are written right on the homework and are the only things written on the homework, along with a small picture. I promise you anything that seems off was not something I did but something weird in the assignment from my professor.

I suspect your prof had a couple of nice Clarets before writing this problem up for you ... anyway, if the force is proportional to velocity then that's called a damper. So do you understand why it belongs in the equation?

 

[SauerKrauter]

Reading up on a damper that seems to make much more sense than what he wrote (friction) so thank you everyone for that. Can anyone help with solving the differential equation? as I do not really know much about dampers ( he hasn't gone over them in class ).

 

[haruspex]

show that Y(t) observes the differential equation Y''(t)+(b/m)y'(t)+(k/m)Y(t) = 0 and
find the general solution using Y(t) = const * e^ikt with k related to m, k, and b.
Ce^ikt[i²a²+(b/m)ia+(k/m)]=0

You correctly (half) overcame the error in the statement of the problem wherein k was used to mean two different things. The statement of the problem is also wrong in giving you the form "ikt", implying that k there is real.
To fix it completely, it should be Y = Ce^wt, where w may be complex. This yields Ce^wt[w²+(b/m)w+(k/m)]=0. Assuming C is nonzero, you can reduce that to a quadratic equation in w. Solve it in terms of b, k, m.

 

[rude man]

Reading up on a damper that seems to make much more sense than what he wrote (friction) so thank you everyone for that. Can anyone help with solving the differential equation? as I do not really know much about dampers ( he hasn't gone over them in class ).

You don't need to know about dampers except that they apply a negative force to the mass proportional to its velocity. Just as a spring applies a negative force = -k*x, the damper applies a negative force = -b*dx/dt. That's all you need to know about dampers.

Your ODE is already set to go. It's a second-order linear, constant-coefficient ODE solvable the way I'm sure you were taught.

(Wind resistance is roughly a damper. Actually a better example than friction ...!).

Note the initial condition Y(0+) = L₀.