Mass is non-uniformly distributed over the rod of length l . Its linear mass density varies linearly with length as λ =kx2 . The position of centre of mass (from lighter end) is given by-

Question

Mass is non-uniformly distributed over the rod of length l . Its linear mass density varies linearly with length as λ =kx2 . The position of centre of mass (from lighter end) is given by- (A) 2 l / 5 (B) 3 / l (C) 3 l / 4 (D) 2 l / 3

Tardigrade Admin · Accepted Answer

Xc m=( displaystyle ∫ x d m/ displaystyle ∫ d m) Xc m=( displaystyle ∫ x λ d x/ displaystyle ∫ λ d x) Xc m=( displaystyle ∫ 01 x (k x2 d x)/ displaystyle ∫ 01 k x2 d x) Xc m=(l4 / 4/l3 / 3)=(3/4)l

Q. Mass is non-uniformly distributed over the rod of length $l$ . Its linear mass density varies linearly with length as $λ = k x^{2}$ . The position of centre of mass (from lighter end) is given by-

$2 l /5$

$3/ l$

$3 l /4$

$2 l /3$

$X_{c m} = \frac{\int x d m}{\int d m}$
$X_{c m} = \frac{\int x λ d x}{\int λ d x}$
$X_{c m} = \frac{\int _{0}^{1} x ( k x ^{2} d x )}{\int _{0}^{1} k x ^{2} d x}$
$X_{c m} = \frac{l ^{4} /4}{l ^{3} /3} = \frac{3}{4} l$

Q. Mass is non-uniformly distributed over the rod of length l . Its linear mass density varies linearly with length as λ=kx2 . The position of centre of mass (from lighter end) is given by-

2l/5

3/l

3l/4

2l/3