The following parameters of the arrangement of figure are available: the angle `alpha` which the inclined plane forms with the horizontal, and the coefficient of friction k between the body `m_1` and the inclined plane. The masses of the pulley and the threads, as well as the friction in the pulley, are negligible. Assuming both bodies to be motionless at the initial moment, find the mass ratio `m_2//m_1` at which the body `m_2`
(a) starts coming down,
(b) starts going up,
(c) is at rest.

Explanation

At the initial moment, obviously the tension in the thread connecting `m_1` and `m_2` equals the weight of `m_2`.
(a) For the block `m_2` to come down or the block `m_1` to go up, the condition is
`m_2g-Tge0` and `T-m_1g sin alpha-frge0`
where `T` is tension and `f_r` is friction which in the limiting case equals `km_1gcosalpha`. Then
or `m_2g-m_1sin alphagtkm_1gcosalpha`
or `(m_2)/(m_1)gt(k cos alpha+sin alpha)`
(b) Similarly in the case
`m_1g sin alpha-m_2g gt f r_(lim)`
or, `m_1sin alpha-m_2ggtkm_1g cos alpha`
or, `m_2/m_1lt(sin alpha-kcos alpha)`
(c) For this case, neither kind of motion is possible, and `f r` need not be limiting.
Hence, `(kcos alpha+sin alpha)gtm_2/m_1gt(sin alpha-k coa alpha)`