Solving Hooke's Law Problems: Compression & Forces

[Sarah0001]

Homework Statement

A uniform beam AOB, O being the mid point of AB, mass M, rests on three identical vertical springs with stiffness constants k1, k2 and k3 at A, O and B respectively. The bases of the springs are fixed to a horizontal platform. Determine the compression of the springs and their compressional forces in the case:

(i) k1 = k3 = k and k2 = 2k
(ii) k1 = k, k2 = 2k and k3 = 3k

2.
The use of Hookes law is only of need here and the rods weight for the first part. The second part additionally requires the use moments which I am confused on as well as the relationship between the relationship between the spring constant and their length of compressions relative to one another.

The Attempt at a Solution

The assumption for the first scenario is that the compression (x) is the same for k2 as it is for k1 and k3. The markscheme makes indication this this is due to the fact that the system of rod and the three springs are symmetrical, hence compression for the middle spring must be the same as the other two.
I can understand this and which I have done.
for part (ii) I am confused the relationship between the length of the compression between the three springs:

Assume spring K3 length compressed by X
K2 " " X + B
K3 " " X+2B

Aswell as the use of moments on this question seen in the uploaded image (solution)

[/B]

 

[haruspex]

It is not entirely clear which parts are your work, which parts are as explained to you, and exactly what your question is.
Are you asking for an explanation of the assumption that the compressions form an arithmetic sequence?

There is a slight awkwardness in the question. If we assume that the springs were vertical before compression then the tilting of the beam will mean they are no longer vertical. The geometry gets rather messy. The solution you attached assumes the bases of the springs move closer together so that they can remain vertical.

What is it that you do not understand about the balance of moments equation? It seems straightforward.

 

[Sarah0001]

sorry I did forget to summarise what I was confused on. Its yes (a) why of the assumption that the compressions form an arithmetic sequence?

(b) The part of the equation I'm stuck on is why resolve using cosine theta , and why does it lead the conclusion 2α = 2β.

 

[haruspex]

why of the assumption that the compressions form an arithmetic sequence?

Because the beam is straight and the attachments are equally spaced. If the beam is length 2L and slooes at angle θ then the centre of the beam must be L sin(θ) below one end and L sin(θ) above the other.

why resolve using cosine theta , and why does it lead the conclusion 2α = 2β.

The magnitude of the torque from a force about an axis is the magnitude of the force multipled by the perpendicular distance to the axis (that is, perpendicular to the line of action of the force).
The forces from the springs are vertical, so the perpendicular distances are the horizontal displacements, L cos(θ).

Taking moments about the centre, the two end springs are both at distance L cos(θ), so that cancels, leaving us to conclude that the two forces are equal: k(α+2β)=3k(α).

 

[Sarah0001]

Thank you!