Answer:
a) ρL(x)= 20+4/5*x
b) the total mass of the rod is M=6000gr = 6 Kg
c) the center of mass of the rod is X= 61.11 cm
Explanation:
The linear mass density ρL is defined as
ρL(x)= dm(x)/dx , where m(x) is the mass of the rod at x (d represents the derivative operator)
a) since the linear density is a linear function:
ρL(x)= a+b*x
at x= 0 → ρL(x)=a =20 g/cm
at x=1 m=100 cm → ρL(x)=ρL(x)= 20 g/cm +b* (100 cm) = 100 g/cm → b= (100 g/cm - 20 g/cm)/100 cm = 4/5 g/cm²
ρL(x)= 20+4/5*x
b) considering that ρL(x)= dm(x)/dx → dm(x) = ρL(x) dx → m(x) = ∫ρL(x) dx + C
m(x) = ∫ρL(x) dx + C = ∫ (20+4/5*x) dx + C = 20x +4/5 (x²/2) + C = 2/5 x² + 20 x + C
now
m(0) = 0 → m(0) = 2/5 (0) + 20 (0) + C = C = 0
therefore
m(x) = 2/5 x² + 20 x
since the rod has 1m=100 cm long
M = m(100 cm) = 2/5 gr/cm² (100cm)² + 20 gr/cm (100cm) = 6000 gr = 6 Kg
c) the center of mass of the rod X is
X = (∫x dm) /M (evaluated between integration limits 0 and 100 cm of length)
but dm(x) = ρL(x) dx
X = (∫x ρL(x) dx ) /M = (1/M)*(∫ x*(20+4/5*x) dx ) = (1/M)*(∫ (20*x+4/5*x²) dx ) =
(1/M) ( 20(x²/2)+ 4/5*(x³/3) ) = (1/6000) ( 20*(100²/2) )+ 4/5*(100³/3) - 0 ) = 61.11 cm
therefore the center of mass X of the rod is X=61.11 cm
