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                 \[{{x}_{CM}}=\frac{1}{M}\,\,\int{x\,dm=0}\]                 \[\frac{1}{M}\,\,\int_{0}^{L}{x\left( \frac{M}{L}\,dx \right)}\]                                 \[=\left[ \frac{1}{L}\frac{{{x}^{2}}}{2} \right]_{0}^{L}=\frac{L}{2}\] They-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 \[\frac{L}{2}\].