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Grade: 12
        A thin bar of length "L" has a mass per unit length "a" that increases linearly with distance from one end. If its total mass is M and its mass per unit length at the lighter end is "b", then the distance of the centre of the mass from lighter end is ?
6 years ago

Answers : (2)

Sanchayan Ghosh
27 Points
							So this is how you do it. 
Imagine that the left end of the rod has a mass per unit length of b, which increases to the right end as a.

Consider a small element dx at a distance x from the lighter end.
let mass per unit length be ?
now, ?  a x.
therefore, ? = kx.
at the heavier end, ? = a.
so, a = kL.
or, k = a/L
Now, centre of mass = 1/M (? xdm).
dm = ?dx.
dm = kxdx.
therefore, xdm = x^2dx
Integrating and taking limits 0 to L. 
? xdm =k ?  x^2dx = x^3 /3
So, simplifying, we get, distance of centre of mass = 1/M (a/L) (L^3/3) = a(L^2/3M).
Since the mass of the rod is given: b becomes unnecessary.
						
6 years ago
Abhijeet
12 Points
							Thanx !
						
6 years ago
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