Hooke's Law & SHM Homework: Solution Attempt

[physics_31415]

Homework Statement

See attachment (titled "Statement.jpg")

Homework Equations

F = ma
F = -kx
U = K = (1/2)kx^2
I'm assuming there are more...

The Attempt at a Solution

My first attempt at this soultion was to use energy methods. The force applied for some time t0 will displace the block by some distance x. I then calculated the potential energy in the spring and used COE. However I eneded up with a useless result. It was something like (x-x0)^2 = (x-x0)^2.

Any hints you can give me will be GREATLY appreciated! I have also attached the free-body diagram that I came up with (titled "FBD.jpg"). Thank you in advance!

 

[Redbelly98]

Can you write an equation of motion valid during the time that F is acting? For this, you may need to figure out the new equilibrium position during that time.

Then, how much work does the force F do during time t0?

p.s. welcome to PF.

 

[physics_31415]

Redbelly98,

Thank you for your response! I took a stab at an equation of motion but am not confident in it. What I've come up with is a driven 2nd-order diff eq. Is this correct? Is it then just a matter of solving for x? I have attached a pdf showing the details (it was quicker than trying to figure out Latex).

Also, you mentioned the work done during the time that the force was "active". Wouldn't this just be F*x where x is the displacment? I'm missing something on how to incorporate t0.

 

[Redbelly98]

Hmmm, I was thinking that we "just know" sinusoidal motion is involved, so no need to set up and solve a differential equation. Just have to figure out equilibrium position and amplitude, and whether it's sine or cosine.

 

[physics_31415]

Redbelly98,

Thank you for your input! I managed to find the solution to this problem. It was by no means as straight forward as I initially thought. It ultimately involved solving a driven 2nd order diff-eq and useing Euler's identity a bunch of times. I suspect there is an easier method (perhaps using energy methods). I'd be happy to share the final result with you if you would like. Thanks!

 

[Redbelly98]

Sure, (if it's not too much trouble) you can post your solution. Glad you were able to solve it.

 

[physics_31415]

Attached is the solution to this problem. Thanks!