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Question Number 57124 by necx1 last updated on 30/Mar/19

Commented by necx1 last updated on 30/Mar/19

$p l e a s e h e l p$
Commented by mr W last updated on 31/Mar/19

$d e n s i t y o f r o d 1 = \frac{M}{L}$ $d e n s i t y o f r o d 2 = \frac{3 M}{3 L} = \frac{M}{L}$ $b o t h r o d s h a v e t h e s a m e d e n s i t y, s o$ $t h e w h o l e r o d a l s o h a s u n i f o r m d e n s i t y .$ $w h o l e r o d :$ $m a s s m = 4 M$ $l e n g t h l = 4 L$ $c e n t e r o f m a s s = i n t h e m i d d l e o f l$ $I = \frac{{m l}^{2}}{12} = \frac{(4 M) {(4 L)}^{2}}{12} = \frac{16 {M L}^{2}}{3}$
Commented by necx1 last updated on 31/Mar/19

$h m m m . . . . T h i s i s r e a l l y s i m p l e t o u n d e r s t a n d .$
	Answered by tanmay.chaudhury50@gmail.com last updated on 31/Mar/19
	$1)[origin (0,0,0) is taken at free end of rod L] 2)co ordinate of joined end of two rod(L,0,0) 3)co ordinate of free end of rod 3L(4L,0,0) 4)co ordinate of centre of mass of rod of lengthL ((L/2),0,0) 5)co ordinate of C.M of rod of length 3L {(L+((3L)/(2 ))),0,0}=(((5L)/2),0,0) co ordinate of centre of mass combined rod is x=((m×(L/2)+3m×((5L)/2))/(m+3m))=((8mL)/(4m))=2L c.m(2L,0,0) Moment of inertia =M×(L^2 /(12))+M(2L−(L/2))^2 +3M×(((3L)^2 )/(12))+3M(2L−((5L)/2))^2 =((ML^2 )/(12))+((M×9L^2 )/4)+((3M×9L^2 )/(12))+((3M×L^2 )/4) =((28ML^2 )/(12))+((12ML^2 )/4) =((28ML^2 +36ML^2 )/(12))=((64ML^2 )/(12))=((16ML^2 )/3) pls check is it correct$
	$1) [o r i g i n (0, 0, 0) i s t a k e n a t f r e e e n d o f r o d L]$ $2) c o o r d i n a t e o f j o i n e d e n d o f t w o r o d (L, 0, 0)$ $3) c o o r d i n a t e o f f r e e e n d o f r o d 3 L (4 L, 0, 0)$ $4) c o o r d i n a t e o f c e n t r e o f m a s s o f r o d o f l e n g t h L$ $(\frac{L}{2}, 0, 0)$ $5) c o o r d i n a t e o f C . M o f r o d o f l e n g t h 3 L$ ${(L + \frac{3 L}{2}), 0, 0} = (\frac{5 L}{2}, 0, 0)$ $c o o r d i n a t e o f c e n t r e o f m a s s c o m b i n e d r o d i s$ $x = \frac{m \times \frac{L}{2} + 3 m \times \frac{5 L}{2}}{m + 3 m} = \frac{8 m L}{4 m} = 2 L$ $c . m (2 L, 0, 0)$ $M o m e n t o f i n e r t i a$ $= M \times \frac{L^{2}}{12} + M {(2 L - \frac{L}{2})}^{2} + 3 M \times \frac{{(3 L)}^{2}}{12} + 3 M {(2 L - \frac{5 L}{2})}^{2}$ $= \frac{{M L}^{2}}{12} + \frac{M \times 9 L^{2}}{4} + \frac{3 M \times 9 L^{2}}{12} + \frac{3 M \times L^{2}}{4}$ $= \frac{28 {M L}^{2}}{12} + \frac{12 {M L}^{2}}{4}$ $= \frac{28 {M L}^{2} + 36 {M L}^{2}}{12} = \frac{64 {M L}^{2}}{12} = \frac{16 {M L}^{2}}{3}$ $p l s c h e c k i s i t c o r r e c t$
	Commented by necx1 last updated on 31/Mar/19

	$i n d e p t h s i r . . . . T h a n k y o u f o r y o u r h e l p .$ $I^{'} v e l e a r n t s o m u c h s i r .$
	Commented by tanmay.chaudhury50@gmail.com last updated on 31/Mar/19

	$t h a n k y o u a l s o t o g o t h r o u g h t h e s o l u t i o n$
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