Rotation 2 Question 22
22. 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 an angular velocity $\omega _0$ about an axis perpendicular to the rod and passing through the mid-point of the rod (see figure).
There are no external forces. When the beads reach the ends of the rod, the angular velocity of the system is…… .
$(1988,2 M)$
True/False
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Answer:
Correct Answer: 22. $\frac{M \omega_0}{M+6 m}$
Solution:
- $I _1 \omega _1=I _2 \omega _2$
$$ \begin{aligned} \therefore \quad \omega _2 & =\frac{I _1}{I _2} \cdot \omega _1=\frac{\left(M L^{2} / 12\right)}{\left[\left(M L^{2} / 12\right)+2 m(L / 2)^{2}\right]} \omega _0 \\ & =\frac{M L^{2}}{M L^{2}+6 m L^{2}} \omega _0=\frac{M}{M+6 m} \omega _0 \end{aligned} $$
