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A sphere of mass m and radius r is pushe...

A sphere of mass `m` and radius `r` is pushed onto the fixed horizontal surface such that it rolls without slipping from the beginning. Determine the minimum speed `v` of its mass centre at the bottom so that it rolls completely around the loop of radius `(R + r)` without leaving the track in between -
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Text Solution

Verified by Experts

The correct Answer is:
`v = sqrt((27)/(7) gR)`
at `A`
`mg + N = (mv^('2))/(R)`
For minimum velocity `N = 0`
`v' = sqrt(gR)` ….(1)
From energy conservation
`(1)/(2) mv^(2)(1 +(K^(2))/(R^(2))) = mg(2R) +(1)/(2) mv^('2) (1+ (K^(2))/(R^(2)))`
`(1)/(2) mv^(2)(1 +(2)/(5)) = mg(2R) +(1)/(2) mv^('2) (1+(2)/(5))`
`(7)/(10) mv^(2) =mg (2R) +(7)/(10) mv^('2)` ....(2)
from (1) & (2)
`(7)/(10) v^(2) = (2gR) + (7)/(10) xx gR`
`(7)/(10) v^(2) = (27 gR)/(10), v = sqrt((27 gR)/(7))`.
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