A red carpet surprisingly found conducting has resistance \(m\) and total mass \(M\) and when rolled completely has radius \(R.\) With a gentle negligible push it starts rolling on the ground with a velocity of \(v\) and radius \(r.\) There exists a uniform magnetic field perpendicular to the base of the carpet the whole time. Let the whole motion stop after a time \(T.\) The rms value of \(vr\) during this time interval from \(t=0\) to \(t=T\) is \(n.\) The centre is connected to the other end by a long wire of negligible resistance. Neglect any strain in the carpet. Then what is the value of \([(n^2)/4]?\)

**Details and Assumptions**

- \(T=10\text{ sec}\)
- \(R=5\text{ m}\)
- \(M=2\text{ kg}\)
- \(m=30\ \Omega\)
- \(g=9.8\text{ m/s}^{2}\)
- \(B=\pi\text{ T}\)
- [.] denotes the greatest integer function

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