Solve for Period of Oscillation of 2 Springs with Mass Attached

[oneplusone]

Homework Statement

You have two springs with spring constants k_1 = 10 and k_2 = 20 vertically attached to a wall, and a mass of mass 3.00kg is hung from it. Find the period of oscillation.


----------------
||
|| <-- first spring, k_1 = 10
||
--
||
|| <== second spring, k_2 = 20
||
{} <== mass 3.00kg

Homework Equations

F=-kx
T = 2pi/omega

The Attempt at a Solution

Consider the second spring. We have:
$F=-kx \implies 3(-9.8) = -(20)(x) \implies x = 1.47$

Now the second spring undergoes a displacement of:

$F=-kx \implies 3(-9.8) = -(10)x \implies x = 2.94$.

So k effective is:

$F = -k (x_1+x_2) \implies 3(9.8) = k(2.94+1.47) \implies k = 4.41$

From here, i just found $\omega$ by using $\omega =\sqrt{k}{m}$. and the period was easy from there.

Is the first part of my solution correct? I am having trouble visualizing it.

 

[Doc Al]

Consider the second spring. We have:
$F=-kx \implies 3(-9.8) = -(20)(x) \implies x = 1.47$

Now the second spring undergoes a displacement of:

$F=-kx \implies 3(-9.8) = -(10)x \implies x = 2.94$.

So far so good.

So k effective is:

$F = -k (x_1+x_2) \implies 3(9.8) = k(2.94+1.47) \implies k = 4.41$

4.41 is the total displacement, not the spring constant. Solve for the effective spring constant.

 

[oneplusone]

oops i meant 6.666. Thank you.

A follow up question to this has the same springs, except oriented like this:

-------------------
|| ||
|| ||
|| ||
|| ||
------
| {} |
------With the springs parallel. How would you solve this? I am still not convinced that there is a solution (to me, the "block" will be tilted).

 

[Doc Al]

With the springs parallel. How would you solve this? I am still not convinced that there is a solution (to me, the "block" will be tilted).

Assume the system is constrained so that the springs get the same displacement.

 

[HalfLostAndSearching]

Yes, your solution is correct. By using the equation F=-kx, you have correctly found the displacements of both springs. The effective spring constant, k effective, is also calculated correctly by considering both springs in series. This value is then used to find the angular frequency, ω, and the period of oscillation can be easily calculated from there. Your solution is also clearly presented and easy to follow. Good job!