The centre of mass of a non-uniform rod of length L whose mass per unit length λ =(K x2/L) , Where k is a constant and x is the distance from one end is:

Question

The centre of mass of a non-uniform rod of length L whose mass per unit length λ =(K x2/L) , Where k is a constant and x is the distance from one end is: (A) (3 L/4) (B) (L/8) (C) (K/L) (D) (3 K/L)

Tardigrade Admin · Accepted Answer

dm=(k x2/L)dx ∴ xc m=( displaystyle ∫ 0L x d m/ displaystyle ∫ 0L d m)=( displaystyle ∫ 0L (k x3/L) d x/ displaystyle ∫ 0L (k x2)L d x)=(((L4/4))/((L3)3))=(3 L/4)

Q. The centre of mass of a non-uniform rod of length $L$ whose mass per unit length $λ = \frac{K x ^{2}}{L}$ , Where k is a constant and $x$ is the distance from one end is:

$\frac{3 L}{4}$

$\frac{L}{8}$

$\frac{K}{L}$

$\frac{3 K}{L}$

$d m = \frac{k x ^{2}}{L} d x$
$∴ x_{c m} = \frac{\int _{0}^{L} x d m}{\int _{0}^{L} d m} = \frac{\int _{0}^{L} \frac{k x ^{3}}{L} d x}{\int _{0}^{L} \frac{k x ^{2}}{L} d x} = \frac{( \frac{L ^{4}}{4} )}{( \frac{L ^{3}}{3} )} = \frac{3 L}{4}$

Q. The centre of mass of a non-uniform rod of length L whose mass per unit length λ=LKx2​ , Where k is a constant and x is the distance from one end is:

43L​

8L​

LK​

L3K​