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A rod of length L is placed along the x-...

A rod of length L is placed along the x-axis between `x=0` and `x=L`. The linear mass density (mass/length) `rho` of the rod varies with the distance x from the origin as `rho=a+bx`. Here, a and b are constants. Find the position of centre of mass of this rod.

Text Solution

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As rod is kept along x - axis
Hence, `Y_(cm)=0` and `Z_(cm)=0`
For x - coordinate
`x_(cm)=(int_(0)^(L)xdm)/(int_(0)^(L)dm)`
`dm=lambda dx = (A+Bx)dx`.
`x_(cm)=(int_(0)^(L)x(A+Bx)dx)/(int_(0)^(L)(A+Bx)dx)=((AL^(2))/(2)+(BL^(3))/(3))/(AL+(BL^(2))/(2))=(L(3A+2BL))/(3(2A+BL))`
Hence, coordinates of the centre of mass are `((L(3A+2BL))/(3(2A+BL)), theta, 0)`
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