A rod of length 50 cm has linear density (mass-per-length) given by l = 50. 0 g/m + 20. 0x g/m2, where x is the distance from one end, measured in meters. The distance of its center of mass from x=0 is 0.257m
The total mass of a linearly distributed mass object is calculated by integrating the linear mass density with respect to the length of the linear mass object. The mass of the rod can be calculated as-
m = ∫(x=0 to x=L) λdx
where L= length of the rod
= 50cm or 0.5m
λ= mass per unit length
= 50.0g/m + 20.0xg/m²
m= ∫(x=0 to x=0.5)λdx
m= ∫(x=0 to x=0.5) (50+20x)dx
= [50x + 20x²/2]⁰.⁵
=50* 0.5 + 10 * (0.5)²
m= 27.5g
xcm = 1/m ∫(x=0 to x= L) xdm
=1/m ∫(x=0 to x= L) xλdx
=1/m ∫(x=0 to x=0.5) x (50+20x)dx
=1/m [50x²/2 + 20x³/3] ⁰.⁵
=1/27.5 [ 25 * (0.5)² + 20/3 (0.5)³]
xcm=0.275m
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