Calculus 3 tension question?

Question

The tension T at each end of the chain has magnitude 35 N. What is the weight of the chain? Give your answer correct to two decimal places. (now each end is 37 degrees)

Answer 1

I'm only familiar with the particle model, where we consider all forces going through a single point, so I'm not sure if this is correct...

Consider the center of the chain to be the point through which all forces go through.

-------------------------------

.....\.37°...........37°/  

......\............... ../  

........\............. /  

..........\......... /  

............\..... /  

..............\../  

...............O

................|  

................|  

................|  

................|  

As we can see from the above diagram, the triangle formed is an isosceles triangle. Therefore, the third angle in the triangle is equal to...

180° - 37° - 37° = 106°

Because the tension is equal through the string, we expect the downward force to bisect the third angle, 106°. Therefore, the other two angles around the center 'O' are equal to...

(360° - 106°) / 2 = 127°

We know that the downward force should equal to...

F = m*g

... because of the downward pulling force of gravity (note: 'm' represents the weight of the chain)

-------------------------------

..35N..\....37°.....37°../   35N

...........\................../  

.............\.............. /  

...............\........... /  

.................\.106°./  

...................\...../  

...........127°....O....127°

......................|  

......................|  

......................|  

......................|  

...................m*g N

Applying Lami's theorem;

35 / sin (127°) = (m*g) / sin (106°)

Taking g= ~ 9.8

We obtain the answer; m= 4.30 (to 2 dp)

Therefore the mass of the chain is 4.30 kg

[excuse the shoddy diagrams.... I couldn't think of a better way to represent and explain the question]