Question

If the linear density (mass per unit length) of a rod of length `3 m` is proportional to `x`, where `x`, where `x` is the distance from one end of the rod, the distance of the centre of gravity of the rod from this end is.
A. 2.5 m
B. 1 m
C. 1.5 m
D. 2 m

Answer 1

Correct Answer - D
(d) Consider an element of length `dx` at a distance `x` from end `A`.
Here, mass per unit length `lamda` of rod
`lamda prop x rArr lamda = kx`
:. `dm = lamda dx = kxdx`
Position of centre of gravity of rod from end `A`.
`x_(CG) = (int_0^L xdm)/(int_0^L dm)`
`x_(CG) = (int_0^3 x(kx dx))/(int_0^3 kx dx) =[(x^3)/(3)]_0^3/[[(x^2)/(2)]_0^3) = ((3)^3/(3))/((3)^3/(2)) = 2 m`.
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