Question

Q. A rod of uniform thickness is placed along x-axis with one end at origin . If length of rod is l and its linear mass density is proportional to x, then find distance of its centre of mass from origin.

Answer 1

Dear Student ,
$$\begin{gathered} Linearmassdensity\lambda =kx \\ wherekisthecons\tan t \\ atx=L,\lambda =\frac{M}{L} \\ k=\frac{M}{L^2} \\ letdmisthesmallmasselementatdis\tan cexfromorigin. \\ dm=\lambda x=\frac{M}{L^2}xdx \\ Dis\tan ceofthecentreofmassisgivenby \\ R=\frac{1}{M}\int xdm \\ R=\frac{1}{M}{\int }_0^{L}xdm \\ R=\frac{1}{M}{\int }_0^{L}x\frac{M}{L^2}xdx \\ R=\frac{L}{3} \end{gathered}$$

Regards