Question

if the linear density of a rod of length L varies as ^=kx3/L where k is a constant and x is the distance of any point from one end ,then find the distance of centre of mass from the end at x=0​

Answer 1

Heya........!!!!


It is Given that => λ = x + 2 .

=> Considering a small part dx of the Rod and then it's mass will be dm .

=> Then it's mass dm => λ × dx = ( x + 2 )dx .

=> let Xc denotes the centre of Gravity

Then through Integrating , , we get ,,

=>> Xc = ( ∫  xdm ÷  ∫ dm )
=> Xc =  ∫  l xdx ÷  ∫ l dx (( here l = lamda ))

=> Putting the value of lamba we get ;

➡Xc =  ∫ x(x + 2 ) dx ÷  ∫ ( x + 2 ) dx

((( Note** = We are integrating with limit of Lenght ( L ) to 0 )))

On solving that Integral and Put L = 3m as it is given in the question .

♦♦Finally we get =>> Xc = 12/7 m



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