question_answer

A straight rod of length L has one of its ends at the origin and the other at x - L. If the mass per unit length of the rod is given by A x, where A is constant, where is its mass centre?            

A)  \[\frac{L}{3}\]                                  

B)        \[\frac{L}{2}\]

C)         \[\frac{2L}{3}\]                                

D)         \[\frac{3L}{4}\]

Correct Answer: B

Solution :

Let the mass of an element of length \[dx\]of rod located at distance \[x\]away from left end is\[\frac{M}{L}dx.\]The \[x-\]coordinate of the centre of mass is given by \[{{x}_{CM}}=\frac{1}{M}\int_{{}}^{{}}{x\,dm=0}\] \[=\frac{1}{M}\int_{0}^{L}{x\left( \frac{M}{L}dx \right)}\] \[=\left[ \frac{1}{L}\frac{{{x}^{2}}}{2} \right]_{0}^{L}=\frac{L}{2}\] The \[y-\]coordinate is \[{{Y}_{CM}}=\frac{1}{M}\int_{{}}^{{}}{y\,dm}=0\] and similarly, \[{{z}_{CM}}=0\] The centre of the mass is at\[\left( \frac{L}{2},0,0 \right)\] or at the middle point of the rod, i.e., at\[\frac{L}{2}.\]