Question

Where is the center of mass, $x$?

• A. $\frac{3\ell }{5}$
• B. $\frac{6\ell }{5}$
• C. $\frac{4\ell }{5}$
• D. $\frac{7\ell }{5}$

Answer 1

The correct option is C $\frac{4\ell }{5}$

We have the center of mass as
$\overline{x}=\frac{{\int }_0^{l}xdm}{{\int }_0^{l}dm}$
or
$\overline{x}=\frac{{\int }_0^{l}x\rho Adx}{{\int }_0^l\rho Adx}$
or
$\overline{x}=\frac{{\int }_0^{l}x{\rho }_o(\frac{x}{l}{)}^{3}dx}{{\int }_0^l{\rho }_o(\frac{x}{l}{)}^{3}dx}$
or
$\overline{x}=\frac{{\int }_0^l{x}^{4}dx}{{\int }_0^l{x}^{3}dx}$
or
$\overline{x}=\frac{\frac{l^5}{5}}{\frac{l^4}{4}}$
or
$\overline{x}=\frac{4l}{5}$