Question

Point masses 1, 2, 3 and 4 kg are lying at the points \[(0,0,0)\], \[(2,0,0)\] , \[(0,3,0)\] and \[( - 2, - 2,0)\] respectively. The moment of inertia of this system about X-axis will be:
A. \[43kg{m^2}\]
B. \[34kg{m^2}\]
C. \[27kg{m^2}\]
D. \[72kg{m^2}\]

Answer 1

Hint:The moment of inertia of a particle of mass M due to rotation about an axis at a perpendicular distance r from it and parallel to it is \[M{r^2}\]. This perpendicular distance can be called the radius of the circle which is the rotation path of the particle. For a system of n particles with masses \[{M_1},{M_{2,}}.........,{M_n}\] and radii \[{r_1},{r_2},..........,{r_n}\], the moment of inertia of the entire system is the sum of moment of inertia of each particle.

Formula Used:
The moment of inertia of particle at a perpendicular distance from the axis of rotation is given by:
\[i = M{r^2}\] ---- (1)
Where i = moment of inertia of the particle
M = mass of the particle
r = perpendicular distance from the axis of rotation
Moment of inertia for a system of n such particles is given by:
\[I = \sum\limits_{k = 1}^n {{i_k}} = \sum\limits_{k = 1}^n {{M_k}{r_k}^2} \]----- (2)

Complete step by step solution:
Given: Four particles of masses 1 , 2 , 3 , 4 kg at the points \[(0,0,0)\], \[(2,0,0)\] , \[(0,3,0)\] and \[( - 2, - 2,0)\]on the cartesian coordinate system.
Moment of inertia of each particle about the axis of rotation (x-axis) is given by,
\[i = M{r^2}\]
The moment of inertia of the system about an axis of rotation is the sum of moment of inertia of all the particles about the axis of rotation.

Thus, the total moment of inertia of the given system of four particles about the x-axis is:
 \[I = \sum\limits_{k = 1}^n {{i_k}} = \sum\limits_{k = 1}^n {{M_k}{r_k}^2} \].
Here k = 4.So,
\[I = \sum\limits_{k = 1}^4 {{M_k}{r_k}^2} = 1 \times {(0)^2} + 2 \times {(0)^2} + 3 \times {(3)^2} + 4 \times {( - 2)^2}\]
\[\therefore I = 43\,kg{m^2}\]

Hence option A is the correct answer.

Note: If we have to calculate the moment of inertia about an axis parallel to the axis passing through the centre of the object, we use the theorem of parallel axis. When the body has symmetry about two out of three axes in a 3D plane, perpendicular axis theorem is used. We have to calculate the perpendicular distance between the particle and axis of rotation to determine the moment of inertia of a complete system of particles. If all the particles of the system of masses have the same mass, then we also measure the radius of gyration or gyradius which is the root mean square distance of the particles from the axis of rotation. It is applicable to rigid rotator type cases.