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Q.

Three identical uniform thin metal rods each of mass “m” and length “L” are arranged in the XY plane as shown in the figure. A fourth uniform thin rod of mass “3m” is placed as shown in the figure in the XY plane. The value of length of the fourth rod such that the centre of mass of all the four rods lies at the origin is

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a

$\frac{L (2 \sqrt{2} + 1)}{2}$

b

$\frac{L (\sqrt{2} + 1)}{3}$

c

2L

d

3L

answer is C.

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Detailed Solution

$(\frac{l}{2} \cos 45, \frac{L}{2} \sin 45)$

$(\frac{L}{2 \sqrt{2}}, \frac{L}{2 \sqrt{2}})$

Question Image

$(- l, \cos 45 ° - l, \sin 45 °)$

$= (- L, \sqrt{2}, - L_{1} \sqrt{2})$

$x_{c m} = \frac{\frac{m L}{2} + \frac{m L}{2 \sqrt{2}} + 0 - \frac{3 m L_{1}}{\sqrt{2}}}{m + m + m + 3 m}$

$∴ 0 = \frac{L}{2} + \frac{L}{2 \sqrt{2}} - \frac{3 l_{1}}{\sqrt{2}}$

$2 l_{1} = \frac{L}{3} (\sqrt{2} + 1)$

Length of 3m mass rod

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Three identical uniform thin metal rods each of mass “m” and length “L” are arranged in the XY plane as shown in the figure. A fourth uniform thin rod of mass “3m” is placed as shown in the figure in the XY plane. The value of length of the fourth rod such that the centre of mass of all the four rods lies at the origin is