Question

A uniform thin rod ab of length l has linear mass density = a+bx/l where x is measured from a if the cm pf the rod lies at a distance of 7l/12 from a then a and b are related as

Answer 1

Hello dear,

● Answer- 2a = b

● Explaination-
Here, all distances are measured from point mass a.

Centre of mass is calculated by -
X(CM) = ∫ux.dx / ∫u.dx
7l/12 = ∫(a + bx/l)x.dx / ∫(a + bx/l).dx
7l/12 = ∫(ax + bx^2/l).dx / ∫(a + bx/l).dx
7l/12 = (bl^3/3l + al^2/2) / (al + bl^2/2)
7l/12 = (al/2 + bl^2/3) / (al + bl^2/3)

Solving this, you'll get
2a = b

Hope this helps you..

Answer 2

density=a+bx/l

Xcm=7l/12

int(a+bx/l)x.dx÷int(a+bx/l).dx=7l/12

by whole integration

3al^2+2bl^2÷6al+3bl=7l/12

42a+21b=36a+24b

6a=3b

2a=b