Question

Four particle of mass ${\text{1g, 2g, 3g, and 4g}}$ are at ${\text{1cm, 2cm, 3cm, and 4cm}}$ from the axis of rotation respectively then the moment of inertia of the system in $g\,c{m^2}$ is
(A) 63
(B) 100
(C) 36
(D) 48

Answer 1

Hint: In the question they have provided the mass of the particles and the distance of the particles from the axis of rotation. We have to find the moment of inertia of the system.
Complete step by step answer
Formula used: ${\text{I}} = m \times {r^2}$
Moment of inertia is also known as angular mass or rotational inertia. It is the torque required for an angular acceleration about an axis of rotation. It is represented by I. The moment of inertia is specified with a chosen axis of rotation. It depends on the distribution of mass around the axis of rotation i.e. the moment of inertia varies depending on the axis of rotation. Basically, moment of inertia is used to calculate the angular momentum.
In other words it is the product of mass of particle in the axis of rotation and the square of distance from the axis of rotation. Its unit is $Kg\,{m^2}$
Which is,
${\text{I}} = m \times {r^2}$
Where,
$I$ is the moment of inertia.
$m$ is the sum of product of all the mass.
$r$ is the distance from the axis of rotation.
Given,
The mass of the particles are ${\text{1g, 2g, 3g, and 4g}}$ and the distance between the particles and the axis of rotation is ${\text{1cm, 2cm, 3cm, and 4cm}}$ respectively
${\text{I}} = m \times {r^2}$
The moment of inertia of a system is the sum of the product of mass of every particle in the axis of rotation and the square of distance from the axis of rotation.
${\text{I = }}\sum {{m_i}{r_i}^2} $
$ \Rightarrow {\text{I = (1}} \times {{\text{1}}^2}{\text{) + (2}} \times {{\text{2}}^2}{\text{) + (3}} \times {{\text{3}}^2}{\text{) + (4}} \times {4^2})$
$ \Rightarrow {\text{I = (1}} \times {\text{1) + (2}} \times 4{\text{) + (3}} \times 9{\text{) + (4}} \times 16)$
$ \Rightarrow {\text{I = (1) + (8) + (27) + (64}})$
$ \Rightarrow {\text{I = 100 g c}}{{\text{m}}^2}$

Hence the correct answer is option (B) ${\text{100}}\,{\text{g}}\,{\text{c}}{{\text{m}}^{\text{2}}}$

Note: Moment of inertia formula differs based on the systems, whether it is a continuous system or a discrete system. Here in the question they have mentioned to write the unit in $g\,c{m^2}$ so don’t change the unit to standard unit.