Angular frequency of a mass between two springs.

[Aesteus]

Homework Statement

A mass M is suspended from the ceiling by a spring with spring constant k, and from the floor by a spring with spring constant 3k. Find the frequency of the mass' oscillation.

Homework Equations

F=ma

The Attempt at a Solution

F(net) = Mg + kx - 3kx = Mg - 2kx

performing a variable switch z= x - Mg/2k, I simplified the equation and set it equal to F(net) = Ma.

Ma = -2kz ... so therefore ω = (2k/M)^1/2

As you see, I performed a variable switch and solved the angular frequency for z. Now, how do I go about switching back to x?

 

[ehild]

F=ma

The Attempt at a Solution

F(net) = Mg + kx - 3kx = Mg - 2kx

performing a variable switch z= x - Mg/2k, I simplified the equation and set it equal to F(net) = Ma.

Ma = -2kz ... so therefore ω = (2k/M)^1/2

As you see, I performed a variable switch and solved the angular frequency for z. Now, how do I go about switching back to x?

z= x - Mg/2k. Is not x=z+Mg/2k?

ehild

 

[Aesteus]

:) very good

The problem is that I'm trying to switch back to x from my z-based frequency equation ω=(k/M)^1/2. And how do I do that? .... Or is there another way?

 

[ehild]

The frequency is the same either you solve for x or z. z is a simple harmonic motion: z=Asin(ωt), with ω=sqrt(2k/M). x is an SHM + constant, but ω does not change.

If you do not like to use the new variable, try to find the solution directly for x of form x(t)=Asin(ωt)+B, (B is a undefined constant, you get if you substitute x into the original equation).

Physically, the springs change length with respect to their unstretched length when the mass is placed between them, and the mass will vibrate around that equilibrium point where the forces from the spring cancel with gravity. What is the change of length of both springs?

ehild

 

[Aesteus]

Ah I see now. And I think part of the problem is that it's 5 a.m. here. :)
Also, do you think you can help me out with my other frequency problem? It's about finding angular frequency from potential energy. I've hit a mental wall.

https://www.physicsforums.com/showthread.php?t=593279

 

[ehild]

Ah I see now. And I think part of the problem is that it's 5 a.m. here. :)

Go to sleep. You will figure out the solution in your dreams

ehild