question_answer

A chain of mass M and length l is suspended vertically with its lower end touching a weighing scale. The chain is released and falls freely onto the scale. Neglecting the size of r the individual links, what is the reading of the scale when a length x of the chain has fallen?  

A) \[\frac{Mgx}{l}\]           

B) \[\frac{2Mgx}{l}\]

C) \[\frac{3Mgx}{l}\]         

D) \[\frac{4Mgx}{l}\]

Correct Answer: C

Solution :

[c]: Mass per unit length, \[\lambda =\frac{M}{l}.\] The descending part of the chain is in free fall, also its every point has descended by a distance x.   So, speed of each point, \[v=\sqrt{2gx}\]               Assume a very small distance dx falls in a short internal of time dt.     Normal exerted on the falling part,      \[N=-\frac{d{{p}_{x}}}{dt}=\frac{-(0-(\lambda dx)v)}{dt}\] \[=\lambda {{v}^{2}}=\lambda (2gx)=2\lambda gx\] Normal due to x part of the chain   on the weighing machine, \[N'=\lambda gx\] Reading of the scale \[W=N+N'=3\lambda gx\]