First slide

Methods to locate centre of mass

Question

A uniform disc of diameter R/2 is put over another uniform disc of diameter R of same thickness and density. The peripheries of the two discs touch each other. The centre of mass of the system from centre of big disc is

Moderate

A

R/10
B

R/20
C

9R/10
D

11R/10

Solution

For disc $\large m\propto R^2$
$\large {m_1} \propto {\left( {\frac{R}{2}} \right)^2}\, \Rightarrow \,{m_1} = \frac{{{R^2}}}{4}.k$
$\large {m_2} \propto {\left( {\frac{R}{4}} \right)^2}\, \Rightarrow \,{m_1} = \frac{{{R^2}}}{16}.k$
$\large d = \frac{R}{2} - \frac{R}{4} = \frac{R}{4}$
distance of CM from big disc
$\large = \frac{{{m_2}d}}{{{m_1} + {m_2}}} = \frac{{\frac{{{R^2}}}{{16}}\left( {\frac{R}{4}} \right)}}{{\frac{{{R^2}}}{4} + \frac{{{R^2}}}{{16}}}} = \frac{R}{{20}}$

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