A rod of length L is placed along the X-axis between x = 0 and x … - Vidyadip

Question

A rod of length $L$ is placed along the $X-$axis between $x = 0$ and $x = L$. The linear density $($mass$/$ length$) \rho$ of the rod varies with the distance x from the origin as $\rho=\text{a + bx.}$

Find the $SI$ units of $a$ and $b.$
Find the mass of the rod in terms of $a, b$ and $L.$

✓

Answer

$\rho=\frac{\text{mass}}{\text{length}}=\text{a + bx}$

$S.I$. unit of $‘a’ = \ kg/m$ and $SI$ unit of $‘b’ = kg/m^2 ($from principle of homogeneity of dimensions$)$
Let us consider a small element of length $‘dx\ ’$ at a distance $x$ from the origin as shown in the figure.

$\therefore\ dm =$ mass of the element $=\rho\text{ dx = (a + bx)dx}$ So, mass of the rod $= m$ $\int\text{dm}=\int\limits^{\text{L}}_0(\text{a + bx)dx}$
$=\Big[\text{ax}+\frac{\text{bx}^2}{2}\Big]^{\text{L}}_0=\text{aL}+\frac{\text{bL}^2}{2}$

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