Two particles of masses m1 and m2 are connected by a rigid...

Two particles of masses $m_{1}$ and $m_{2}$ are connected by a rigid massless rod of length $r$ to constitute a dumb-bell which is free to move in the plane. The moment of inertia of the dumb-bell about an axis perpendicular to the plane passing through the centre of mass is (a) $\frac{m_{1} m r^{2}}{m_{1}+m_{1}}$ (b) $\left(m_{1}+m_{1}\right) r^{3}$ (c) $\frac{m_{1} m_{t} r^{2}}{m_{1}-m_{2}}$ (d) $\left(m_{1}-m_{2}\right) r^{2}$

Two particles of masses $m_{1}$ and $m_{2}$ are connected by a rigid massless rod of length $r$ to constitute a dumb-bell which is free to move in the plane. The moment of inertia of the dumb-bell about an axis perpendicular to the plane passing through the centre of mass is
(a) $\frac{m_{1} m r^{2}}{m_{1}+m_{1}}$
(b) $\left(m_{1}+m_{1}\right) r^{3}$
(c) $\frac{m_{1} m_{t} r^{2}}{m_{1}-m_{2}}$
(d) $\left(m_{1}-m_{2}\right) r^{2}$

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Key Concepts

Rigid Body Systems in Two-Particle Models

In a rigid body system formed by two masses connected by a massless rod, the relative positions of the masses are fixed. This allows for the computation of the overall moment of inertia by determining each mass's distance from the center of mass, and then applying the definition of the moment of inertia through the summation of each mass multiplied by the square of its distance from the rotation axis.

Moment of Inertia

This is a measure of an object's resistance to changes in its rotational motion about an axis. In the context of a rigid body, such as a system made up of two masses connected by a rod, it is calculated by summing the products of each mass and the square of its distance from the axis of rotation.

Center of Mass

This is the weighted average of the positions of all the masses in a system. For a two-particle system, the center of mass is the point at which the mass distribution can be considered to be concentrated, and it serves as the axis for analyzing the rotational motion of the system.

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