Question

The linear density of a rod of length 3m varies as $\lambda =2+x$, then the position of the centre of gravity of the rod is
A. $\dfrac{7}{12}m$
B. $\dfrac{5}{13}m$
C. $\dfrac{12}{7}m$
D. $\dfrac{14}{4}m$

Answer 1

Hint: We need to take the help of integration to solve this problem. Consider an infinitesimal mass at a certain distance. Find the value of the mass using the given relation. Multiply the mass with the distance from one end, and integrate this quantity over the length. Divide it with the total mass to find the centre of gravity.

Complete step by step answer:

Let’s look at the diagram:

Let’s assume there is an infinitesimal element of length dx.

We can use the given equation to find the mass of that element.

Hence, the mass of the element is,
$dm=\lambda .dx$

The centre of gravity can be given by,
${{x}_{cm}}=\dfrac{\int{x.dm}}{\int{dm}}$

Where,
$dm$ is the mass of the infinitesimal element
$x$ is the distance from one side.

We can calculate both separately.

So, we can write,
$\int\limits_{0}^{3}{x.dm}=\int\limits_{0}^{3}{x.(2+x)dx}$
$=\int\limits_{0}^{3}{(2x+{{x}^{2}})dx}$
$=[{{x}^{2}}+\dfrac{{{x}^{3}}}{3}]_{0}^{3}$
$=9+\dfrac{27}{3}$
$=9+9=18$

We can again write,
$\int\limits_{0}^{3}{dm}=\int\limits_{0}^{3}{(2+x)dx}$
$=[2x+\dfrac{{{x}^{2}}}{2}]_{0}^{3}$
$=2\times 3+\dfrac{9}{2}=\dfrac{21}{2}$

So, the centre of gravity is given by,
${{x}_{cm}}=\dfrac{\int{x.dm}}{\int{dm}}$
${{x}_{cm}}=\dfrac{18}{(\dfrac{21}{2})}=\dfrac{36}{21}=\dfrac{12}{7}$

Hence, the correct option is (C).

Note: Centre of gravity is the location in which we can assume that the entire mass is concentrated. If we calculate the total mass of the object, and replace the object with a point mass at the centre of gravity.

Centre of gravity should not necessarily be on the object. For example, a ring has the centre of gravity at the centre of the ring. Hence, the centre of gravity can be outside the object as well.
The procedure for calculating the centre of gravity for an object having variable mass remains like this. You need to calculate the mass of an infinitesimal element and integrate with respect to it.