# Period of the pulley-spring system

## Homework Statement

We have the system consisting of a massless pulley and two massless springs as the picture shown below. Two springs has spring constants as $/k_1$ and $/k_2$. The object has mass m

Let's calculate the period of the vibration of the object if it is disturbed vertically in a small distant from the equibrilium

## The Attempt at a Solution

on progress

Simon Bridge
Homework Helper
Great - get back to us when you've got stuck ;)

My lecturer give the problem without giving a result, and so I got stuck of how to deal with it. If someone know the solution, or at least, the exact result, please let me know

Simon Bridge
Homework Helper
You are training to be able to cope with problems which have no known solution ... you cannot expect to always be told the answers in advance.

... you have been learning some physics in class and you have a bunch of lecture notes: use what you've learned. Give it a go and we can help where you get stuck. To do that, we need to see your reasoning and your working.

my friend show me result http://codecogs.izyba.com/png.latex?\%20\varpi%20=%20\sqrt{%20\frac{k_1%20k_2}{m(4k_2%20+%20k_1)}%20} [Broken]
, it seems to be true but I still do not know the solution

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Simon Bridge
Homework Helper
I'm sorry, nobody can help you if you don't tell us your reasoning and what you have tried.

I am trying to analyze nature in this prob as follows
Consider this system in equibrilium and then, disturb the object a small distance $x$ vertically
and call corresponding distances of spring $k_1$ and $k_2$ as $x_1$ and $x_2$
then we obtain the relation between these variables

$\left\{ \matrix{ k_1 x_1 + k_2 x_2 = m\ddot x \cr x = x_1 + x_2 \cr} \right.$

though I obtain two relations above but these are still not rich enough to resolve, help me

Simon Bridge
Homework Helper
Well no, you have not used all your knowledge of the system.
It is difficult to tell what you need to do exactly because you have not yet shown me your working.

I'll go quickly - you will need to check:

we displace the mass a small distance x
this moves the pulley, and pulls on the springs.

for the mass: ##T-mg=m\ddot{x}## ...(1)

by symmetry, the T on the mass is also the T on spring 2 forcing an extension x2:

##T=k_2x_2## ...(2)

The tension forcing spring 1 is not going to be the same so I'll call it T1

##T_1=k_1x_1## ...(3)

All the tensions meet at the pulley ... here I'm not certain because I think this should be unbalanced but I think it comes out in the wash if I write:

##T_1=2T## ...(4)

Looking at the way the cable loops over the pulley - if spring 1 drops by x1, the pulley drops that far too, which makes x1 additional cord available on both sides to contribute to x - in addition to any extension on x2 ... so:

##x=2x_1+x_2## ...(5)

Count them up - that's five simultaneous equations and five unknowns.
Sub (4) -> (3) to eleiminate T1, then use (2) and (3) to get relations ofr x1 and x2 which you can put into (5). Solve (1) for T and put that into (5) also and you are done.

I have $$Kx-m\ddot{x}=mg\; : \; \frac{1}{K}=\frac{4}{k_1}+\frac{1}{k_2}$$

I am making no guarantees that these are the right ones - it is your work: you have to check. But they should show you where you need to look for additional relations.

even i had a problem with this while trying it using the energy method.