A smooth uniform rod of length L and mass M has two identical beads of negligible size, each of mass m, which can slide freely along the rod. Initially, the two beads are at the centre of the rod and the system is rotating with angular velocity ω$$0$$, about an axis perpendicular to rod and passing through the $$mid-point$$ of rod. There are no external forces. Whenthe beads reach the ends of the rod, the angular velocity of the system is

Question

A smooth uniform rod of length L and mass M has two identical beads of negligible size, each of mass m, which can slide freely along the rod. Initially, the two beads are at the centre of the rod and the system is rotating with angular velocity ω$$0$$, about an axis perpendicular to rod and passing through the $$mid-point$$ of rod. There are no external forces. Whenthe beads reach the ends of the rod, the angular velocity of the system is

Infinity Learn · Accepted Answer

Suppose, when the beads reach the ends, the moment of inertia of the system changes from $$I1$$ to $$I2$$, . As a result, the angular velocity changes from ω$$1to$$ ω$$2$$ there is no external torque acting on the system. Hence, the angular momentum of the system is conserved.$$I1$$ω$$1=I2$$ω$$2ML212+0$$ω$$1=ML212+2mL22$$ω$$2Solving$$, we getω$$2=MM+6m$$ω$$0$$ ∵ ω$$1=$$ω$$0$$

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