A chain of mass per unit length `lambda` and length 1.5 m rests on a fixed smooth sphere of radius `R = (2)/(pi) m` such that A end of chain is at the top of sphere while the other end is hanging freely. Chain is held stationary by a horizontal thread PA. The tension in the thread is:

Options

A. `lambda g ((1)/(2) + (2)/(pi))`
B. `lambda g((pi)/(2) + (2)/(pi))`
C. `lambda g ((2)/(pi))`
D. None of these

Answer

A

Explanation

Length of the hanging part `= (1.5 - (2 pi R)/(4)) = 0.5 m`
`:. T_(1) = 0.5 lambda g`
Consider the force diagram of a small mass `dm = R lambda d theta` which subtends angle `d theta` at the centre and whose position vector with respect to the centre of the sphere makes an angle `theta` with horizontal.
`implies T + dT = T +(dm) g cos theta, N = (dm) g sin theta`
`implies int_(T_(1))^(T_(2)) dT = R lambda g int_(0)^(pi//2) cos theta. d theta`
`implies T_(2) = T_(1) + R lambda g = 0.5 lambda g + (2)/(pi) lambda g = lambda g ((1)/(2) + (2)/(pi))`