22. A thin cylindrical rod is rotating on a horizontal plane about its center of mass, which
is at rest. It breaks instantaneously into two cylindrical halves. The two resulting pieces
a. Move apart without spinning.
b. Spin but have their centres of mass at rest.
c. Move apart and spin in the same direction as the original rod.
d. Move apart and spin in opposite directions.
A thin cylindrical rod rotates on a horizontal plane about its stationary center of mass (CM) and breaks instantly into two identical cylindrical halves. The correct answer is option d: Move apart and spin in opposite directions.
Core Physics Principles
No external forces act on the system, so the overall CM remains at rest post-break. Linear momentum conservation requires the two halves (each mass m/2) to move in opposite directions with equal speeds v, typically tangential to the original rotation. No external torques about the original CM conserve total angular momentum, originally Iω=112mL2ω (z-direction).
Option Analysis
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a. Move apart without spinning: Incorrect. Each half retains rotational motion from the original velocity distribution; pure translation violates angular momentum conservation.
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b. Spin but have their centres of mass at rest: Incorrect. Halves acquire linear velocities at their CMs (originally at ±L/4), causing separation; CMs cannot stay fixed.
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c. Move apart and spin in the same direction as the original rod: Incorrect. Symmetric break induces opposite spins to conserve angular momentum vector; same-direction spin would mismatch total L.
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d. Move apart and spin in opposite directions: Correct. Right half CM moves outward with tangential v=ω(L/4); its orbital angular momentum (m/2)(L/4)v aligns with original, while spin ω′ opposes it. Left half mirrors oppositely. Total: 2[112(m/2)(L/2)2ω′+(m/2)(L/4)2ω]=112mL2ω solves consistently [python].
