Free-hanging string with 4 attached masses

[Nikitin]

Homework Statement

http://home.phys.ntnu.no/brukdef/undervisning/tfy4145/ovinger/Ov03.pdf
look at "oppgave 5".

In the problem, you have a massless string with 4 uniform masses attached symmetrically, and you are supposed to show that the implicit equation for x (and x = cos(alpha)) is correct. Each of the masses are L/5 apart and the distance between the two ends of the string is D.

The Attempt at a Solution

I should be getting 5 equations for the 5 unkowns, but I am only getting 4 (the last 4 equations written on the paper)..

https://fbcdn-sphotos-h-a.akamaihd.net/hphotos-ak-prn2/1234158_10201385459335100_1607600845_n.jpg

 

[SteamKing]

This problem would make a lot more sense if I read Norwegian.

 

[Nikitin]

Look at the picture in the pdf: You see a massless string weighed down by 4 attached point-masses. Each of the mass is a distance L/5 apart from each-other, where L = total length of the string. D= distance between the two ends of the string.

Next you can see two angles: alpha and beta. The point with this problem is to show that the implicit equation for cosine of alpha written in the assignment,where x = cos(a), is correct.

Please tell me if there is something more you do not understand! I really need help with this

 

[grzz]

It is more convenient to simplify your notation by calling the tensions T and S.

 

[grzz]

Then use your D equation to get cosβ in terms of x where x = cosα.

 

[grzz]

I think that you do not need to use S_{3}.

 

[Nikitin]

You are probably right, but if I removed S_3 I would have 4 unkowns and 3 equations.. I don't understand how I can get my last equation?

Also sorry about the notation, but I took it from the assignment.

 

[grzz]

You have FOUR equations.

Two from vertical equilibrium and one from horizontal equilibirum if you do not use the third tension. But you have also your D equation from which to get cosβ in terms of x where x = cosα.

 

[Nikitin]

You mean those involving G? But G is also an unknown, because I don't know the mass. Do you think G would disappear if I inserted the equations into the one for D?

thanks for all help :)

 

[Nikitin]

OK after a fresh look at the problem I see you were completely right. thanks !