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- Homework Statement
- A rigid horizontal smooth rod $AB$ of mass $0.75 kg$ and length $40 cm$ can rotate freely about a fixed vertical axis through its mid point $O$. Two rings each of mass $1 Kg$ are initially at rest at a distance of $10 cm$ from $O$ on the either side of the rod.The rod is set in rotation with an angular velocity of $30$ radians per second. Find the velocity of each ring along the length of the rod in m/s when they reach the ends of the rod.

- Relevant Equations
- (I am new to this forum, I don't know what to write in 'Relevant Equations' field)

Method 1: Simply conserving angular momentum about the the fixed vertical axis and conserving energy gives ##v=3##, which is correct according to my book.

Method 2: Conserving angular momentum when the two rings reach distance ##x## from the centre gives

##(0.01+2x^2) \omega =0.9##

Also in the rod's frame ##a=v dv/dx =\omega ^2 x## (where a and v radial acceleration and velocities).

So, ##v^2/2=\int_{0.1}^{0.2} \frac {0.9xdx} {(0.01+2x^2)^2}=5##.

So, ##v=\sqrt{10}##.

What is wrong in second method?

Edit: Everything is in SI unit

Method 2: Conserving angular momentum when the two rings reach distance ##x## from the centre gives

##(0.01+2x^2) \omega =0.9##

Also in the rod's frame ##a=v dv/dx =\omega ^2 x## (where a and v radial acceleration and velocities).

So, ##v^2/2=\int_{0.1}^{0.2} \frac {0.9xdx} {(0.01+2x^2)^2}=5##.

So, ##v=\sqrt{10}##.

What is wrong in second method?

Edit: Everything is in SI unit

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