QJEEM20Phccd4: JEE Main 2020 Physics - Class 11 JEEM20Phccd4 Solution: A rod of length L has non-uniform linear mass density given by…

Question

Verified Solution • 2025–26 Session moderate

A rod of length L has non-uniform linear mass
density given by

\rho

(x) =

a + b{\left( {{x \over L}} \right)^2}

, where a
and b are constants and 0

\le

x

\le

L. The value
of x for the centre of mass of the rod is at :

Accepted Answer

\rho

= a + b

{\left( {{x \over L}} \right)^2}

dm =

\rho

dx =

\left( {a + b{{{x^2}} \over {{L^2}}}} \right)dx

M =

\int {dm}

=

\int\limits_0^L {\left( {a + b{{{x^2}} \over {{L^2}}}} \right)dx}

X_com =

{{\int {xdm} } \over {\int {dm} }}

=

{{\int\limits_0^L {\left( {a + b{{{x^2}} \over {{L^2}}}} \right)xdx} } \over {\int\limits_0^L {\left( {a + b{{{x^2}} \over {{L^2}}}} \right)dx} }}

=

{{{{a{L^2}} \over 2} + {b \over {{L^2}}}.{{{L^4}} \over 4}} \over {aL + {b \over {{L^2}}}.{{{L^3}} \over 3}}}

=

{{\left( {{{4a + 2a} \over 8}} \right)L} \over {\left( {{{3a + b} \over 3}} \right)}}

=

{3 \over 4}\left( {{{2a + b} \over {3a + b}}} \right)L

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