Q) The density of a linear rod of length L varies as ρ = A + Bx, where - Physics - Current Electricity

Question

Q) The density of a linear rod of length L varies as ρ = A +
Bx, where x is the distance from the left end Locate the centre of
mass
X C M = 3 A L + 2 B L 2 3 2 A + B L

Pintu B. · Accepted Answer

Dear Student ,

The linear mass density is defined as the ratio of mass over its
length that is,
λ=Mx

let us consider small mass of the rod of small length
dx,
therefore the small mass will be given in differential form
as,
dm=λdx

on substituting the values of linear mass density of the rod we
get,
dm=A+Bxdx

on integrating the above equation for length ranging from 0 to L we
get the total mass of the rod as,
∫0Mdm=∫0LA+BxdxM=∫0LAdx+∫0LBxdx

on integrating and on substituting the limits we
get,
M=Ax0L+Bx20L2M=AL+BL22

now we know that the centre of mass of a rigid body is given
as,
c m =1M∫0Lλxdx

on substituting for M and λ 
we get,
c m =1AL+BL22∫0LA+Bxxdxc m
=1AL+BL22∫0LAxdx+∫0LBx2dx

on integrating and on substituting the limits we
get,
c m =1AL+BL22Ax220L+Bx330Lc m =1AL+BL22AL22+BL33c m
=LAL2+BL23LA+BL2c m =AL2+BL23A+BL2c m =3AL+2BL26A+3BL

hence centre of mass of the rod will be at c m
=3AL+2BL26A+3BL

Regards

Q). The density of a linear rod of length L varies as ρ = A + Bx, where x is the distance from the left end. Locate the centre of mass. X C M = 3 A L + 2 B L 2 3 2 A + B L

Q). The density of a linear rod of length L varies as $ρ$ = A + Bx, where x is the distance from the left end. Locate the centre of mass.
$[X_{C M} = \frac{3 A L + 2 B L^{2}}{3 (2 A + B L)}]$