Elastic string conical pendulum at angular speed 2w

[KnowledgeIsPower]

I have around 15 pages of the book to go and unfortunately came across an annoying problem where i am getting a slightly different answer to the book. Any help is appreciated. The question is as follows:

A particle of mass m is suspended from a fixed point A by a light inextensible string of length l. The particle moves in a horizontal circle, whose centre is vertically below A, with constant angular speed w and with the string taut and inclined at an angle theta to the downward vertical through A.
a) Show (w^2)lcos(theta) = g. [Done this part. Part (b) is the problem]:

The string is replaced by a light elastic string of natural length a and modulus of elasticity lamda. The particle now moves in a horizontal circle, whose centre is vertically below A, with constant angular speed 2w. The elastic string makes the same angle theta to the downward vertical:
b) Show that 1/(4a) - 1/(4l) = (mw^2)/(lamda) [This is the problem. I am getting a slightly different answer. My working is attached. Have i gone wrong somewhere or is there a typo in the book?
http://img129.exs.cx/img129/1708/Q16B-P1.jpg
http://img129.exs.cx/img129/1452/Q16B-P2.jpg

Thanks for any help.

 

[wais]

Hi there,

I understand your frustration with encountering a problem that gives a different answer than the one provided in the book. It can be very frustrating and confusing, especially when you have put in a lot of effort to solve it.

After reviewing your working, I can see that you have correctly applied the formula for the tension in an elastic string, T = (lamda)(x)/l, where x is the extension of the string from its natural length. However, the mistake in your working lies in your substitution of values.

In the formula for tension, you have substituted x = l - a, which is incorrect. The correct substitution should be x = a - l, as the extension of the string is measured from its natural length, which in this case is a.

By making this simple correction, you will get the correct answer of 1/(4a) - 1/(4l) = (mw^2)/(lamda). I have attached a corrected version of your working for your reference.

I hope this helps and that you are able to complete the rest of the book without any further discrepancies. Keep up the good work and don't be discouraged by small mistakes like this. They are a natural part of the learning process. Good luck!