Question

If mass is a function of $x$, $m = 3x$. Find the COM of rod length $1\;{\text{m}}$.

Answer 1

In this question, we calculate the centre of mass of the rod by using integration method. To understand the centre of mass of the rod, consider the diagram of the rod. And suppose a particle of mass $dm$ and the length of the particle $dx$.

Complete step by step answer:
In this question, we have given a rod of length $1\;{\text{m}}$ and the mass is the function of $x$ that is $m = 3x$. In this problem, we need to calculate the center of mass of the rod.
As we know that the center of mass of a point where the whole body’s mass is supposed to be located or concentrated.
Let us consider a rod of length $L$ and take the mass of rod $m$.

Let us take the small element on the uniform rod which is the length $dx$ and take the mass on the rod is $dm$.
The centre of mass may be anywhere at the rod. The center of mass is having only the $X -$coordinate.
Represent the $X -$coordinate of center of mass by ${X_{CM}}$.
Now we can write the center of mass on ${X_{CM}}$ as,
$\Rightarrow {X_{CM}} = \dfrac{{\int\limits_0^L {xdm} }}{{\int\limits_0^L {dm} }}......\left( 1 \right)$
As we know that the mass of the rod is given as,
$\Rightarrow m = 3x$
Now we differentiate the above equation and get,
$\Rightarrow dm = 3dx$
Now, we substitute the value in equation (1),
$\Rightarrow {X_{CM}} = \dfrac{{\int\limits_0^L {x\left( {3dx} \right)} }}{{\int\limits_0^L {\left( {3dx} \right)} }}$
After integrating above equation, we get,
$\Rightarrow {X_{CM}} = \dfrac{{\left| {\dfrac{3}{2}{x^2}} \right|_0^1}}{{\left| {3x} \right|_0^1}}$
After applying the limits, we get
$\Rightarrow {X_{CM}} = \dfrac{{\dfrac{3}{2}\left[ {{{\left( 1 \right)}^2} - {0^2}} \right]}}{{3 - 0}}$
Now, we simplify the above expression and get,
$\therefore {X_{CM}} = \dfrac{1}{2}\;{\text{m}}$
Therefore, the centre of the mass of the rod is at $\dfrac{1}{2}\;{\text{m}}$.

Note: As we know that the center of mass of a particle is the position of the system. Center of mass is the average position of the particle of the system. Here, it lies in the middle of the rod which is not true for all the cases.