question_answer

A straight rod 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 Ay where A is constant, where is its mass centre?

A)  \[L/3\]                                

B)  \[L/2\]

C)   \[2L/3\]                             

D)  \[3L/4\]

Correct Answer: B

Solution :

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