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) $\frac{m_{1} m_{2} r^{2}}{\left(m_{1}+m_{2}\right)}$ (b) $\left(m_{1}+m_{2}\right) r^{2}$ (c) $\frac{m_{1} m_{2} r^{2}}{\left(m_{1}-m_{2}\right)}$ (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) $\frac{m_{1} m_{2} r^{2}}{\left(m_{1}+m_{2}\right)}$
(b) $\left(m_{1}+m_{2}\right) r^{2}$
(c) $\frac{m_{1} m_{2} r^{2}}{\left(m_{1}-m_{2}\right)}$
(d) $\left(m_{1}-m_{2}\right) r^{2}$

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Key Concepts

Moment of Inertia

Moment of inertia is a physical quantity that measures the resistance of an object to changes in its rotational motion about a specified axis. In the context of a composite system, such as a pair of particles, it is calculated by summing the products of each mass with the square of its distance from the axis of rotation. This concept is central to analyzing rotating systems in mechanics.

Center of Mass

The center of mass of a system is the unique point where the weighted relative position of the distributed mass sums to zero. For a multi-particle system, it is found by taking a mass-weighted average of all particle positions. When analyzing the rotational dynamics of a system, choosing the center of mass as the axis of rotation often simplifies the calculation of the moment of inertia.

Reduced Mass

Reduced mass is a useful concept in two-body problems that simplifies the analysis of the system by combining the two masses into a single effective mass. It is defined as the product of the two masses divided by their sum. This concept plays a role in determining the moment of inertia for systems where the distribution of mass is separated by a fixed distance.

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