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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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Mass of element PQ of length dx situated at x = x is dm =`rho dx =(a+bx) dx`
The CM of the element has coordinates (x, 0,0).
Therefore, x - coordinate of COM of the rod will be
`x_(CM) =(int_(0)^(L)x dm)/(int_(0)^(L)dm) =(int_(0)^(L)(x)(a+bx)dx)/(int_(0)^(L)(a+bx)dx) `
or `x_(CM) =(3aL + 2bL^(2))/(6a + 3bl)`
The y -coordinate of CM of the rod is `y_(CM) =(intydm)/(int dm) =0` (as y=0)
Similarly, `z_(CM) = 0`
Hence, the centre of mass of the rod lies at `[(3aL + 2bL^(2))/(6a + 3bL),0,0]`
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