parvy
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A light elastic string of modulus Lambda and natural length 3a is fixed to two points on the same horizontal level distance 4a apart. Two particles of weight W are attached one at each point of trisection. If the sloping string make an angle @ with the horizontal in equilibrium prove (1) the central portion of the string extends by (a/3) (3 - [email protected])
(2) [email protected] = 3W/Lambda(3- [email protected])

================================ ========================
I have had an attempt but I think I'm heading in the wrong direction


<-------- 4a ---------------->
-----------|----------|-----------
4a/3 4a/3 4a/3
using Hookes Law T = La/3a = L/3 (L = lambda --> modulus of elaticity)

With weight attached we have:

b b
................................ .....
* @ | *
* | *
* | *
* | *
*********************
|<-- l=4a/3 + y --> |
W W
let cental portion string lenth l = 4a/3 + y ( y is the extension from unweighted string)
let the sloping part extend by x ==> sloping part len = 4a/3 + x
hence y = l - 4a/3
but l = 4a - [email protected] = 4a - 2(4a/3 + x)[email protected]
==> y = 8a/3 - 2(4a/3 + x)[email protected]

................................ ...........
* @ ^ *
* | [email protected] *
T * | *
* | *
<[email protected] *****>T****
| |
W W
Let T be the tension in the string
Energy stored in elastic with no weights attahced = La^2/2*3a = La/6
Energy stored in sloping part of string (weights attached) = L3x^2/8a
Energy stored in central portion of string (weights attached) = L3y^2/8a

by conservation of mechanical energy:
[email protected] no weights = Potential Energy + E of sloping part + E of central portion
==> La/6 = -W(4a/3 + x)[email protected] + L3x^2/8a + L3y^2/8a
==> La/6 = 3L/8a(x^2 + y^2) - W(4a/3 + x)[email protected]
...things are getting complicated now, so i beleive my approach is wrong,
Please help!!
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KittyRe-play
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Is this Physics A level?
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parvy
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Hi KittyRe-play
Yes it is part of the mechanics course in maths and/or physics at A level

I'm interested in mechanics and am self teaching at home

..... looks as if my diagrams have been skewed through the editor, never mind i think someone will still be able to make sense of the question!!
Thanks
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