Two identical spherical balls of mass $M$ and radius $R$ each are stuck on two ends of a rod of length $2R$ and mass $M$ (see figure). The moment of inertia of the system about the axis passing perpendicularly through the centre of the rod is :
- $\frac{152}{15} MR^{2}$
- $\frac{17}{15} MR^{2}$
- $\frac{137}{15} MR^{2}$
- $\frac{209}{15} MR^{2}$
The Correct Option is C
Solution and Explanation
$I_{\text{ball}} = \frac{2}{5} MR^{2} +M\left(2R\right)^{2} $
$ =\frac{22}{5} MR^{2}$
$2$ Balls so $ \frac{44}{5}MR^{2} $
Irod = for rod $ \frac{M\left(2R\right)^{2}}{R} =\frac{MR^{2}}{3} $
$ I_{\text{system}} =I_{\text{Ball}} +I_{\text{rod}} $
$ = \frac{44}{5} MR^{2} + \frac{MR^{2}}{3} $
$ = \frac{137}{15} MR^{2} $
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Concepts Used:
Moment of Inertia
Moment of inertia is defined as the quantity expressed by the body resisting angular acceleration which is the sum of the product of the mass of every particle with its square of a distance from the axis of rotation.
Moment of inertia mainly depends on the following three factors:
- The density of the material
- Shape and size of the body
- Axis of rotation
Formula:
In general form, the moment of inertia can be expressed as,
I = m × r²
Where,
I = Moment of inertia.
m = sum of the product of the mass.
r = distance from the axis of the rotation.
M¹ L² T° is the dimensional formula of the moment of inertia.
The equation for moment of inertia is given by,
I = I = ∑mi ri²
Methods to calculate Moment of Inertia:
To calculate the moment of inertia, we use two important theorems-
- Perpendicular axis theorem
- Parallel axis theorem