Question

Two bodies of masses \[1kg\] and \[2kg\] are attached to the ends of a \[2m\] long weightless rod. This system is rotating about an axis passing through the middle point of rod and perpendicular to length. Calculate the M.I. of system.

Answer 1

Hint: Calculate the moment of inertia of the system by assuming that the system is a combination of two particles and a rod attaching them.
We can use the definition of the moment of inertia of a system which states that the moment of inertia of a system is equal to the sum of the product of mass with the square of corresponding distance from the line of action.

Complete step by step answer:
According to the question, we have two bodies of masses \[1kg\] and \[2kg\]. These masses are attached to the ends of a weightless rod of length \[2m\]. This system is rotating about an axis as shown in the figure.

Now, we know that the moment of inertia of this type of system having $n$ particles in the system will be given as-
$M.I. = \sum\limits_n {{m_n}r_n^2} $
Or we can write above equation as-
$M.I. = {m_1}r_1^2 + {m_2}r_2^2 + ....... + {m_n}r_n^2$
Now, in this system, we have two particles in the system. So, the above equation can be deduced in the following equation for a two particle system-
$M.I. = {m_1}r_1^2 + {m_2}r_2^2$ (i)
As the system (rod with two particles) is rotating about an axis which is passing through the midpoint of the rod. So,
${r_1} = {r_2} = 1m$
And from the question, we have ${m_1} = 1kg,{m_2} = 2kg$
So, putting the values of \[{m_1},{m_2},{r_1}\] and \[{r_2}\] in equation (i), we get-
$
  M.I. = 1 \times {(1)^2} + 2 \times {(1)^2} \\
   \Rightarrow M.I. = 1 + 2 \\
   \Rightarrow M.I. = 3kg{m^2} \\
$

Hence, the moment of inertia of this system (rod with two particles) is $3kg{m^2}$.

Note: Remember that the system is rotating about an axis which passes through the centre of the rod. So, the corresponding distance of the particles from a ${r_1} = {r_2} = 1m$. We have to remember the formula of moment of inertia $M.I. = \sum\limits_n {{m_n}r_n^2} $ to solve this question.