Let the length of a rod be 10 meters and the linear density of the rod ρ(x) be written in the

Question

form ρ(x) = ax + b with x = 0 representing the left end of the rod and x = 10 representing
the right end of the rod. If the density of the rod is 2kg/m at the left end and 17 kg/m at the
right end, find the mass and the center of mass of the rod.

Answer 1

ρ(x) = ax + b
ρ(0) = a(0) + b = 2
Thus b = 2 and we have:
ρ(x) = ax + 2
Now,
ρ(10) = a(10) + 2 = 17
Thus a = 1.5 and we have:
ρ(x) = 1.5x + 2
Mass:
M = ∫ (1.5x + 2)dx from 0 to 10
= [0.75x^2 + 2x] from 0 to 10
= [0.75(10)^2 + 2(10) - 0]
= ____ kg

Centre of mass = (1/M) ∫ (x(1.5x + 2)dx from 0 to 10
= (1/____) ∫ (1.5x^2 + 2x)dx from 0 to 10
....you work out the rest.